4 Motion in Two and Three Dimensions
4 Chapter Review
Key Terms
- acceleration vector
- instantaneous acceleration found by taking the derivative of the velocity function with respect to time in unit vector notation
- angular frequency
- ω,ω, rate of change of an angle with which an object that is moving on a circular path
- centripetal acceleration
- component of acceleration of an object moving in a circle that is directed radially inward toward the center of the circle
- displacement vector
- vector from the initial position to a final position on a trajectory of a particle
- position vector
- vector from the origin of a chosen coordinate system to the position of a particle in two- or three-dimensional space
- projectile motion
- motion of an object subject only to the acceleration of gravity
- range
- maximum horizontal distance a projectile travels
- reference frame
- coordinate system in which the position, velocity, and acceleration of an object at rest or moving is measured
- relative velocity
- velocity of an object as observed from a particular reference frame, or the velocity of one reference frame with respect to another reference frame
- tangential acceleration
- magnitude of which is the time rate of change of speed. Its direction is tangent to the circle.
- time of flight
- elapsed time a projectile is in the air
- total acceleration
- vector sum of centripetal and tangential accelerations
- trajectory
- path of a projectile through the air
- velocity vector
- vector that gives the instantaneous speed and direction of a particle; tangent to the trajectory
Key Equations
| Position vector | [latex]\mathbf{\overset{\to }{r}}(t)=x(t)\mathbf{\hat{i}}+y(t)\mathbf{\hat{j}}+z(t)\mathbf{\hat{k}}[/latex] |
| Displacement vector | [latex]\Delta \mathbf{\overset{\to }{r}}=\mathbf{\overset{\to }{r}}({t}_{2})-\mathbf{\overset{\to }{r}}({t}_{1})[/latex] |
| Velocity vector | [latex]\mathbf{\overset{\to }{v}}(t)=\underset{\Delta t\to 0}{\text{lim}}\frac{\mathbf{\overset{\to }{r}}(t+\Delta t)-\mathbf{\overset{\to }{r}}(t)}{\Delta t}=\frac{d\mathbf{\overset{\to }{r}}}{dt}[/latex] |
| Velocity in terms of components | [latex]\mathbf{\overset{\to }{v}}(t)={v}_{x}(t)\mathbf{\hat{i}}+{v}_{y}(t)\mathbf{\hat{j}}+{v}_{z}(t)\mathbf{\hat{k}}[/latex] |
| Velocity components | [latex]{v}_{x}(t)=\frac{dx(t)}{dt}\enspace{v}_{y}(t)=\frac{dy(t)}{dt}\enspace{v}_{z}(t)=\frac{dz(t)}{dt}[/latex] |
| Average velocity | [latex]{\mathbf{\overset{\to }{v}}}_{\text{avg}}=\frac{\mathbf{\overset{\to }{r}}({t}_{2})-\mathbf{\overset{\to }{r}}({t}_{1})}{{t}_{2}-{t}_{1}}[/latex] |
| Instantaneous acceleration | [latex]\mathbf{\overset{\to }{a}}(t)=\underset{t\to 0}{\text{lim}}\frac{\mathbf{\overset{\to }{v}}(t+\Delta t)-\mathbf{\overset{\to }{v}}(t)}{\Delta t}=\frac{d\mathbf{\overset{\to }{v}}(t)}{dt}[/latex] |
| Instantaneous acceleration, component form | [latex]\mathbf{\overset{\to }{a}}(t)=\frac{d{v}_{x}(t)}{dt}\mathbf{\hat{i}}+\frac{d{v}_{y}(t)}{dt}\mathbf{\hat{j}}+\frac{d{v}_{z}(t)}{dt}\mathbf{\hat{k}}[/latex] |
| Instantaneous acceleration as second
derivatives of position |
[latex]\mathbf{\overset{\to }{a}}(t)=\frac{{d}^{2}x(t)}{d{t}^{2}}\mathbf{\hat{i}}+\frac{{d}^{2}y(t)}{d{t}^{2}}\mathbf{\hat{j}}+\frac{{d}^{2}z(t)}{d{t}^{2}}\mathbf{\hat{k}}[/latex] |
| Time of flight | [latex]{T}_{\text{tof}}=\frac{2({v}_{0}\text{sin}\theta )}{g}[/latex] |
| Trajectory | [latex]y=(\text{tan}{\theta }_{0})x-[\frac{g}{2{({v}_{0}\text{cos}{\theta }_{0})}^{2}}]{x}^{2}[/latex] |
| Range | [latex]R=\frac{{v}_{0}^{2}\text{sin}\,2{\theta }_{0}}{g}[/latex] |
| Centripetal acceleration | [latex]{a}_{\text{C}}=\frac{{v}^{2}}{r}[/latex] |
| Position vector, uniform circular motion | [latex]\mathbf{\overset{\to }{r}}(t)=A\text{cos}\,\omega t\,\mathbf{\hat{i}}+A\,\text{sin}\,\omega t\mathbf{\hat{j}}[/latex] |
| Velocity vector, uniform circular motion | [latex]\mathbf{\overset{\to }{v}}(t)=\frac{d\mathbf{\overset{\to }{r}}(t)}{dt}=\text{−}A\omega \,\text{sin}\,\omega t\mathbf{\hat{i}}+A\omega \,\text{cos}\,\omega t\mathbf{\hat{j}}[/latex] |
| Acceleration vector, uniform circular motion | [latex]\mathbf{\overset{\to }{a}}(t)=\frac{d\mathbf{\overset{\to }{v}}(t)}{dt}=\text{−}A{\omega }^{2}\,\text{cos}\,\omega t\mathbf{\hat{i}}-A{\omega }^{2}\,\text{sin}\,\omega t\mathbf{\hat{j}}[/latex] |
| Tangential acceleration | [latex]{a}_{\text{T}}=\frac{d|\mathbf{\overset{\to }{v}}|}{dt}[/latex] |
| Total acceleration | [latex]\mathbf{\overset{\to }{a}}={\mathbf{\overset{\to }{a}}}_{\text{C}}+{\mathbf{\overset{\to }{a}}}_{\text{T}}[/latex] |
| Position vector in frame S is the position vector in frame [latex]{S}^{\prime }[/latex] plus the vector from the origin of S to the origin of [latex]{S}^{\prime }[/latex] |
[latex]{\mathbf{\overset{\to }{r}}}_{PS}={\mathbf{\overset{\to }{r}}}_{P{S}^{\prime }}+{\mathbf{\overset{\to }{r}}}_{{S}^{\prime }S}[/latex] |
| Relative velocity equation connecting two reference frames |
[latex]{\mathbf{\overset{\to }{v}}}_{PS}={\mathbf{\overset{\to }{v}}}_{P{S}^{\prime }}+{\mathbf{\overset{\to }{v}}}_{{S}^{\prime }S}[/latex] |
| Relative velocity equation connecting more than two reference frames |
[latex]{\mathbf{\overset{\to }{v}}}_{PC}={\mathbf{\overset{\to }{v}}}_{PA}+{\mathbf{\overset{\to }{v}}}_{AB}+{\mathbf{\overset{\to }{v}}}_{BC}[/latex] |
| Relative acceleration equation | [latex]{\mathbf{\overset{\to }{a}}}_{PS}={\mathbf{\overset{\to }{a}}}_{P{S}^{\prime }}+{\mathbf{\overset{\to }{a}}}_{{S}^{\prime }S}[/latex] |
Summary
4.1 Displacement and Velocity Vectors
- The position function r⃗ (t)r→(t) gives the position as a function of time of a particle moving in two or three dimensions. Graphically, it is a vector from the origin of a chosen coordinate system to the point where the particle is located at a specific time.
- The displacement vector Δr⃗ Δr→ gives the shortest distance between any two points on the trajectory of a particle in two or three dimensions.
- Instantaneous velocity gives the speed and direction of a particle at a specific time on its trajectory in two or three dimensions, and is a vector in two and three dimensions.
- The velocity vector is tangent to the trajectory of the particle.
- Displacement r⃗ (t)r→(t) can be written as a vector sum of the one-dimensional displacements x⃗ (t),y⃗ (t),z⃗ (t)x→(t),y→(t),z→(t) along the x, y, and zdirections.
- Velocity v⃗ (t)v→(t) can be written as a vector sum of the one-dimensional velocities vx(t),vy(t),vz(t)vx(t),vy(t),vz(t) along the x, y, and zdirections.
- Motion in any given direction is independent of motion in a perpendicular direction.
4.2 Acceleration Vector
- In two and three dimensions, the acceleration vector can have an arbitrary direction and does not necessarily point along a given component of the velocity.
- The instantaneous acceleration is produced by a change in velocity taken over a very short (infinitesimal) time period. Instantaneous acceleration is a vector in two or three dimensions. It is found by taking the derivative of the velocity function with respect to time.
- In three dimensions, acceleration a⃗ (t)a→(t) can be written as a vector sum of the one-dimensional accelerations ax(t),ay(t),andaz(t)ax(t),ay(t),andaz(t) along the x-, y-, and z-axes.
- The kinematic equations for constant acceleration can be written as the vector sum of the constant acceleration equations in the x, y, and z directions.
4.3 Projectile Motion
- Projectile motion is the motion of an object subject only to the acceleration of gravity, where the acceleration is constant, as near the surface of Earth.
- To solve projectile motion problems, we analyze the motion of the projectile in the horizontal and vertical directions using the one-dimensional kinematic equations for x and y.
- The time of flight of a projectile launched with initial vertical velocity v0yv0y on an even surface is given by
Ttof=2(v0sinθ)g.Ttof=2(v0sinθ)g.
This equation is valid only when the projectile lands at the same elevation from which it was launched.
- The maximum horizontal distance traveled by a projectile is called the range. Again, the equation for range is valid only when the projectile lands at the same elevation from which it was launched.
4.4 Uniform Circular Motion
- Uniform circular motion is motion in a circle at constant speed.
- Centripetal acceleration a⃗ Ca→C is the acceleration a particle must have to follow a circular path. Centripetal acceleration always points toward the center of rotation and has magnitude aC=v2/r.aC=v2/r.
- Nonuniform circular motion occurs when there is tangential acceleration of an object executing circular motion such that the speed of the object is changing. This acceleration is called tangential acceleration a⃗ T.a→T. The magnitude of tangential acceleration is the time rate of change of the magnitude of the velocity. The tangential acceleration vector is tangential to the circle, whereas the centripetal acceleration vector points radially inward toward the center of the circle. The total acceleration is the vector sum of tangential and centripetal accelerations.
- An object executing uniform circular motion can be described with equations of motion. The position vector of the object is r⃗ (t)=Acosωtiˆ+Asinωtjˆ,r→(t)=Acosωti^+Asinωtj^, where A is the magnitude ∣∣r⃗ (t)∣∣,|r→(t)|, which is also the radius of the circle, and ωω is the angular frequency.
4.5 Relative Motion in One and Two Dimensions
- When analyzing motion of an object, the reference frame in terms of position, velocity, and acceleration needs to be specified.
- Relative velocity is the velocity of an object as observed from a particular reference frame, and it varies with the choice of reference frame.
- If S and S′S′ are two reference frames moving relative to each other at a constant velocity, then the velocity of an object relative to S is equal to its velocity relative to S′S′ plus the velocity of S′S′ relative to S.
- If two reference frames are moving relative to each other at a constant velocity, then the accelerations of an object as observed in both reference frames are equal.
Conceptual Questions
4.1 Displacement and Velocity Vectors
What form does the trajectory of a particle have if the distance from any point A to point B is equal to the magnitude of the displacement from A to B?
Give an example of a trajectory in two or three dimensions caused by independent perpendicular motions.
If the instantaneous velocity is zero, what can be said about the slope of the position function?
4.2 Acceleration Vector
If the position function of a particle is a linear function of time, what can be said about its acceleration?
If an object has a constant x-component of the velocity and suddenly experiences an acceleration in the y direction, does the x-component of its velocity change?
If an object has a constant x-component of velocity and suddenly experiences an acceleration at an angle of 70°70° in the xdirection, does the x-component of velocity change?
4.3 Projectile Motion
Answer the following questions for projectile motion on level ground assuming negligible air resistance, with the initial angle being neither 0°0° nor 90°:90°: (a) Is the velocity ever zero? (b) When is the velocity a minimum? A maximum? (c) Can the velocity ever be the same as the initial velocity at a time other than at t = 0? (d) Can the speed ever be the same as the initial speed at a time other than at t = 0?
Answer the following questions for projectile motion on level ground assuming negligible air resistance, with the initial angle being neither 0°0° nor 90°:90°: (a) Is the acceleration ever zero? (b) Is the acceleration ever in the same direction as a component of velocity? (c) Is the acceleration ever opposite in direction to a component of velocity?
A dime is placed at the edge of a table so it hangs over slightly. A quarter is slid horizontally on the table surface perpendicular to the edge and hits the dime head on. Which coin hits the ground first?
4.4 Uniform Circular Motion
Can centripetal acceleration change the speed of a particle undergoing circular motion?
4.5 Relative Motion in One and Two Dimensions
What frame or frames of reference do you use instinctively when driving a car? When flying in a commercial jet?
A basketball player dribbling down the court usually keeps his eyes fixed on the players around him. He is moving fast. Why doesn’t he need to keep his eyes on the ball?
If someone is riding in the back of a pickup truck and throws a softball straight backward, is it possible for the ball to fall straight down as viewed by a person standing at the side of the road? Under what condition would this occur? How would the motion of the ball appear to the person who threw it?
The hat of a jogger running at constant velocity falls off the back of his head. Draw a sketch showing the path of the hat in the jogger’s frame of reference. Draw its path as viewed by a stationary observer. Neglect air resistance.
A clod of dirt falls from the bed of a moving truck. It strikes the ground directly below the end of the truck. (a) What is the direction of its velocity relative to the truck just before it hits? (b) Is this the same as the direction of its velocity relative to ground just before it hits? Explain your answers.
Problems
4.1 Displacement and Velocity Vectors
The coordinates of a particle in a rectangular coordinate system are (1.0, –4.0, 6.0). What is the position vector of the particle?
The position of a particle changes from r⃗ 1=(2.0iˆ+3.0jˆ)cmr→1=(2.0i^+3.0j^)cm to r⃗ 2=(−4.0iˆ+3.0jˆ)cm.r→2=(−4.0i^+3.0j^)cm. What is the particle’s displacement?
The 18th hole at Pebble Beach Golf Course is a dogleg to the left of length 496.0 m. The fairway off the tee is taken to be the x direction. A golfer hits his tee shot a distance of 300.0 m, corresponding to a displacement Δr⃗ 1=300.0miˆ,Δr→1=300.0mi^, and hits his second shot 189.0 m with a displacement Δr⃗ 2=172.0miˆ+80.3mjˆ.Δr→2=172.0mi^+80.3mj^. What is the final displacement of the golf ball from the tee?
A bird flies straight northeast a distance of 95.0 km for 3.0 h. With the x-axis due east and the y-axis due north, what is the displacement in unit vector notation for the bird? What is the average velocity for the trip?
A cyclist rides 5.0 km due east, then 10.0 km 20°20° west of north. From this point she rides 8.0 km due west. What is the final displacement from where the cyclist started?
New York Rangers defenseman Daniel Girardi stands at the goal and passes a hockey puck 20 m and 45°45° from straight down the ice to left wing Chris Kreider waiting at the blue line. Kreider waits for Girardi to reach the blue line and passes the puck directly across the ice to him 10 m away. What is the final displacement of the puck? See the following figure.
The position of a particle is r⃗ (t)=4.0t2iˆ−3.0jˆ+2.0t3kˆm.r→(t)=4.0t2i^−3.0j^+2.0t3k^m. (a) What is the velocity of the particle at 0 s and at 1.01.0 s? (b) What is the average velocity between 0 s and 1.01.0 s?
Clay Matthews, a linebacker for the Green Bay Packers, can reach a speed of 10.0 m/s. At the start of a play, Matthews runs downfield at 45°45° with respect to the 50-yard line and covers 8.0 m in 1 s. He then runs straight down the field at 90°90° with respect to the 50-yard line for 12 m, with an elapsed time of 1.2 s. (a) What is Matthews’ final displacement from the start of the play? (b) What is his average velocity?
The F-35B Lighting II is a short-takeoff and vertical landing fighter jet. If it does a vertical takeoff to 20.00-m height above the ground and then follows a flight path angled at 30°30° with respect to the ground for 20.00 km, what is the final displacement?
4.2 Acceleration Vector
The position of a particle is r⃗ (t)=(3.0t2iˆ+5.0jˆ−6.0tkˆ)m.r→(t)=(3.0t2i^+5.0j^−6.0tk^)m. (a) Determine its velocity and acceleration as functions of time. (b) What are its velocity and acceleration at time t = 0?
A particle’s acceleration is (4.0iˆ+3.0jˆ)m/s2.(4.0i^+3.0j^)m/s2. At t = 0, its position and velocity are zero. (a) What are the particle’s position and velocity as functions of time? (b) Find the equation of the path of the particle. Draw the x- and y-axes and sketch the trajectory of the particle.
A boat leaves the dock at t = 0 and heads out into a lake with an acceleration of 2.0m/s2iˆ.2.0m/s2i^. A strong wind is pushing the boat, giving it an additional velocity of 2.0m/siˆ+1.0m/sjˆ.2.0m/si^+1.0m/sj^. (a) What is the velocity of the boat at t = 10 s? (b) What is the position of the boat at t = 10s? Draw a sketch of the boat’s trajectory and position at t = 10 s, showing the x- and y-axes.
The position of a particle for t > 0 is given by r⃗ (t)=(3.0t2iˆ−7.0t3jˆ−5.0t−2kˆ)m.r→(t)=(3.0t2i^−7.0t3j^−5.0t−2k^)m. (a) What is the velocity as a function of time? (b) What is the acceleration as a function of time? (c) What is the particle’s velocity at t = 2.0 s? (d) What is its speed at t = 1.0 s and t = 3.0 s? (e) What is the average velocity between t = 1.0 s and t = 2.0 s?
The acceleration of a particle is a constant. At t = 0 the velocity of the particle is (10iˆ+20jˆ)m/s.(10i^+20j^)m/s. At t = 4 s the velocity is 10jˆm/s.10j^m/s. (a) What is the particle’s acceleration? (b) How do the position and velocity vary with time? Assume the particle is initially at the origin.
A particle has a position function r⃗ (t)=cos(1.0t)iˆ+sin(1.0t)jˆ+tkˆ,r→(t)=cos(1.0t)i^+sin(1.0t)j^+tk^, where the arguments of the cosine and sine functions are in radians. (a) What is the velocity vector? (b) What is the acceleration vector?
A Lockheed Martin F-35 II Lighting jet takes off from an aircraft carrier with a runway length of 90 m and a takeoff speed 70 m/s at the end of the runway. Jets are catapulted into airspace from the deck of an aircraft carrier with two sources of propulsion: the jet propulsion and the catapult. At the point of leaving the deck of the aircraft carrier, the F-35’s acceleration decreases to a constant acceleration of 5.0m/s25.0m/s2 at 30°30° with respect to the horizontal. (a) What is the initial acceleration of the F-35 on the deck of the aircraft carrier to make it airborne? (b) Write the position and velocity of the F-35 in unit vector notation from the point it leaves the deck of the aircraft carrier. (c) At what altitude is the fighter 5.0 s after it leaves the deck of the aircraft carrier? (d) What is its velocity and speed at this time? (e) How far has it traveled horizontally?
4.3 Projectile Motion
A bullet is shot horizontally from shoulder height (1.5 m) with and initial speed 200 m/s. (a) How much time elapses before the bullet hits the ground? (b) How far does the bullet travel horizontally?
A marble rolls off a tabletop 1.0 m high and hits the floor at a point 3.0 m away from the table’s edge in the horizontal direction. (a) How long is the marble in the air? (b) What is the speed of the marble when it leaves the table’s edge? (c) What is its speed when it hits the floor?
A dart is thrown horizontally at a speed of 10 m/s at the bull’s-eye of a dartboard 2.4 m away, as in the following figure. (a) How far below the intended target does the dart hit? (b) What does your answer tell you about how proficient dart players throw their darts?
An airplane flying horizontally with a speed of 500 km/h at a height of 800 m drops a crate of supplies (see the following figure). If the parachute fails to open, how far in front of the release point does the crate hit the ground?
Suppose the airplane in the preceding problem fires a projectile horizontally in its direction of motion at a speed of 300 m/s relative to the plane. (a) How far in front of the release point does the projectile hit the ground? (b) What is its speed when it hits the ground?
A fastball pitcher can throw a baseball at a speed of 40 m/s (90 mi/h). (a) Assuming the pitcher can release the ball 16.7 m from home plate so the ball is moving horizontally, how long does it take the ball to reach home plate? (b) How far does the ball drop between the pitcher’s hand and home plate?
A projectile is launched at an angle of 30°30° and lands 20 s later at the same height as it was launched. (a) What is the initial speed of the projectile? (b) What is the maximum altitude? (c) What is the range? (d) Calculate the displacement from the point of launch to the position on its trajectory at 15 s.
A basketball player shoots toward a basket 6.1 m away and 3.0 m above the floor. If the ball is released 1.8 m above the floor at an angle of 60°60° above the horizontal, what must the initial speed be if it were to go through the basket?
At a particular instant, a hot air balloon is 100 m in the air and descending at a constant speed of 2.0 m/s. At this exact instant, a girl throws a ball horizontally, relative to herself, with an initial speed of 20 m/s. When she lands, where will she find the ball? Ignore air resistance.
A man on a motorcycle traveling at a uniform speed of 10 m/s throws an empty can straight upward relative to himself with an initial speed of 3.0 m/s. Find the equation of the trajectory as seen by a police officer on the side of the road. Assume the initial position of the can is the point where it is thrown. Ignore air resistance.
An athlete can jump a distance of 8.0 m in the broad jump. What is the maximum distance the athlete can jump on the Moon, where the gravitational acceleration is one-sixth that of Earth?
The maximum horizontal distance a boy can throw a ball is 50 m. Assume he can throw with the same initial speed at all angles. How high does he throw the ball when he throws it straight upward?
A rock is thrown off a cliff at an angle of 53°53° with respect to the horizontal. The cliff is 100 m high. The initial speed of the rock is 30 m/s. (a) How high above the edge of the cliff does the rock rise? (b) How far has it moved horizontally when it is at maximum altitude? (c) How long after the release does it hit the ground? (d) What is the range of the rock? (e) What are the horizontal and vertical positions of the rock relative to the edge of the cliff at t = 2.0 s, t = 4.0 s, and t = 6.0 s?
Trying to escape his pursuers, a secret agent skis off a slope inclined at 30°30° below the horizontal at 60 km/h. To survive and land on the snow 100 m below, he must clear a gorge 60 m wide. Does he make it? Ignore air resistance.
A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40°40° with an initial speed of 20 m/s, how close to the green does she come?
A projectile is shot at a hill, the base of which is 300 m away. The projectile is shot at 60°60° above the horizontal with an initial speed of 75 m/s. The hill can be approximated by a plane sloped at 20°20° to the horizontal. Relative to the coordinate system shown in the following figure, the equation of this straight line is y=(tan20°)x−109.y=(tan20°)x−109. Where on the hill does the projectile land?
An astronaut on Mars kicks a soccer ball at an angle of 45°45° with an initial velocity of 15 m/s. If the acceleration of gravity on Mars is 3.7 m/s, (a) what is the range of the soccer kick on a flat surface? (b) What would be the range of the same kick on the Moon, where gravity is one-sixth that of Earth?
Mike Powell holds the record for the long jump of 8.95 m, established in 1991. If he left the ground at an angle of 15°,15°,what was his initial speed?
MIT’s robot cheetah can jump over obstacles 46 cm high and has speed of 12.0 km/h. (a) If the robot launches itself at an angle of 60°60° at this speed, what is its maximum height? (b) What would the launch angle have to be to reach a height of 46 cm?
Mt. Asama, Japan, is an active volcano. In 2009, an eruption threw solid volcanic rocks that landed 1 km horizontally from the crater. If the volcanic rocks were launched at an angle of 40°40° with respect to the horizontal and landed 900 m below the crater, (a) what would be their initial velocity and (b) what is their time of flight?
Drew Brees of the New Orleans Saints can throw a football 23.0 m/s (50 mph). If he angles the throw at 10°10° from the horizontal, what distance does it go if it is to be caught at the same elevation as it was thrown?
The Lunar Roving Vehicle used in NASA’s late Apollo missions reached an unofficial lunar land speed of 5.0 m/s by astronaut Eugene Cernan. If the rover was moving at this speed on a flat lunar surface and hit a small bump that projected it off the surface at an angle of 20°,20°, how long would it be “airborne” on the Moon?
A soccer goal is 2.44 m high. A player kicks the ball at a distance 10 m from the goal at an angle of 25°.25°. What is the initial speed of the soccer ball?
Olympus Mons on Mars is the largest volcano in the solar system, at a height of 25 km and with a radius of 312 km. If you are standing on the summit, with what initial velocity would you have to fire a projectile from a cannon horizontally to clear the volcano and land on the surface of Mars? Note that Mars has an acceleration of gravity of 3.7m/s2.3.7m/s2.
In 1999, Robbie Knievel was the first to jump the Grand Canyon on a motorcycle. At a narrow part of the canyon (69.0 m wide) and traveling 35.8 m/s off the takeoff ramp, he reached the other side. What was his launch angle?
You throw a baseball at an initial speed of 15.0 m/s at an angle of 30°30° with respect to the horizontal. What would the ball’s initial speed have to be at 30°30° on a planet that has twice the acceleration of gravity as Earth to achieve the same range? Consider launch and impact on a horizontal surface.
Aaron Rogers throws a football at 20.0 m/s to his wide receiver, who runs straight down the field at 9.4 m/s for 20.0 m. If Aaron throws the football when the wide receiver has reached 10.0 m, what angle does Aaron have to launch the ball so the receiver catches it at the 20.0 m mark?
4.4 Uniform Circular Motion
A flywheel is rotating at 30 rev/s. What is the total angle, in radians, through which a point on the flywheel rotates in 40 s?
A particle travels in a circle of radius 10 m at a constant speed of 20 m/s. What is the magnitude of the acceleration?
Cam Newton of the Carolina Panthers throws a perfect football spiral at 8.0 rev/s. The radius of a pro football is 8.5 cm at the middle of the short side. What is the centripetal acceleration of the laces on the football?
A fairground ride spins its occupants inside a flying saucer-shaped container. If the horizontal circular path the riders follow has an 8.00-m radius, at how many revolutions per minute are the riders subjected to a centripetal acceleration equal to that of gravity?
A runner taking part in the 200-m dash must run around the end of a track that has a circular arc with a radius of curvature of 30.0 m. The runner starts the race at a constant speed. If she completes the 200-m dash in 23.2 s and runs at constant speed throughout the race, what is her centripetal acceleration as she runs the curved portion of the track?
An experimental jet rocket travels around Earth along its equator just above its surface. At what speed must the jet travel if the magnitude of its acceleration is g?
A fan is rotating at a constant 360.0 rev/min. What is the magnitude of the acceleration of a point on one of its blades 10.0 cm from the axis of rotation?
A point located on the second hand of a large clock has a radial acceleration of 0.1cm/s2.0.1cm/s2. How far is the point from the axis of rotation of the second hand?
4.5 Relative Motion in One and Two Dimensions
The coordinate axes of the reference frame S′S′ remain parallel to those of S, as S′S′ moves away from S at a constant velocity v⃗ SS′=(4.0iˆ+3.0jˆ+5.0kˆ)m/s.v→S′S=(4.0i^+3.0j^+5.0k^)m/s. (a) If at time t = 0 the origins coincide, what is the position of the origin O′O′ in the Sframe as a function of time? (b) How is particle position for r⃗ (t)r→(t) and r⃗ ′(t),r→′(t), as measured in S and S′,S′, respectively, related? (c) What is the relationship between particle velocities v⃗ (t)andv⃗ ′(t)?v→(t)andv→′(t)? (d) How are accelerations a⃗ (t)anda⃗ ′(t)a→(t)anda→′(t) related?
The coordinate axes of the reference frame S′S′ remain parallel to those of S, as S′S′ moves away from S at a constant velocity v⃗ S′S=(1.0iˆ+2.0jˆ+3.0kˆ)tm/sv→S′S=(1.0i^+2.0j^+3.0k^)tm/s. (a) If at time t = 0 the origins coincide, what is the position of origin O′O′ in the Sframe as a function of time? (b) How is particle position for r⃗ (t)r→(t) and r⃗ ′(t)r→′(t), as measured in S and S′,S′, respectively, related? (c) What is the relationship between particle velocities v⃗ (t)andv⃗ ′(t)?v→(t)andv→′(t)? (d) How are accelerations a⃗ (t) anda⃗ ′(t)a→(t) anda→′(t) related?
The velocity of a particle in reference frame A is (2.0iˆ+3.0jˆ)m/s.(2.0i^+3.0j^)m/s. The velocity of reference frame A with respect to reference frame B is 4.0kˆm/s,4.0k^m/s, and the velocity of reference frame B with respect to C is 2.0jˆm/s.2.0j^m/s. What is the velocity of the particle in reference frame C?
Raindrops fall vertically at 4.5 m/s relative to the earth. What does an observer in a car moving at 22.0 m/s in a straight line measure as the velocity of the raindrops?
A seagull can fly at a velocity of 9.00 m/s in still air. (a) If it takes the bird 20.0 min to travel 6.00 km straight into an oncoming wind, what is the velocity of the wind? (b) If the bird turns around and flies with the wind, how long will it take the bird to return 6.00 km?
A ship sets sail from Rotterdam, heading due north at 7.00 m/s relative to the water. The local ocean current is 1.50 m/s in a direction 40.0°40.0° north of east. What is the velocity of the ship relative to Earth?
A boat can be rowed at 8.0 km/h in still water. (a) How much time is required to row 1.5 km downstream in a river moving 3.0 km/h relative to the shore? (b) How much time is required for the return trip? (c) In what direction must the boat be aimed to row straight across the river? (d) Suppose the river is 0.8 km wide. What is the velocity of the boat with respect to Earth and how much time is required to get to the opposite shore? (e) Suppose, instead, the boat is aimed straight across the river. How much time is required to get across and how far downstream is the boat when it reaches the opposite shore?
A small plane flies at 200 km/h in still air. If the wind blows directly out of the west at 50 km/h, (a) in what direction must the pilot head her plane to move directly north across land and (b) how long does it take her to reach a point 300 km directly north of her starting point?
A cyclist traveling southeast along a road at 15 km/h feels a wind blowing from the southwest at 25 km/h. To a stationary observer, what are the speed and direction of the wind?
A river is moving east at 4 m/s. A boat starts from the dock heading 30°30° north of west at 7 m/s. If the river is 1800 m wide, (a) what is the velocity of the boat with respect to Earth and (b) how long does it take the boat to cross the river?
Additional Problems
A Formula One race car is traveling at 89.0 m/s along a straight track enters a turn on the race track with radius of curvature of 200.0 m. What centripetal acceleration must the car have to stay on the track?
A particle travels in a circular orbit of radius 10 m. Its speed is changing at a rate of 15.0m/s215.0m/s2 at an instant when its speed is 40.0 m/s. What is the magnitude of the acceleration of the particle?
The driver of a car moving at 90.0 km/h presses down on the brake as the car enters a circular curve of radius 150.0 m. If the speed of the car is decreasing at a rate of 9.0 km/h each second, what is the magnitude of the acceleration of the car at the instant its speed is 60.0 km/h?
A race car entering the curved part of the track at the Daytona 500 drops its speed from 85.0 m/s to 80.0 m/s in 2.0 s. If the radius of the curved part of the track is 316.0 m, calculate the total acceleration of the race car at the beginning and ending of reduction of speed.
An elephant is located on Earth’s surface at a latitude λ.λ. Calculate the centripetal acceleration of the elephant resulting from the rotation of Earth around its polar axis. Express your answer in terms of λ,λ, the radius RERE of Earth, and time T for one rotation of Earth. Compare your answer with g for λ=40°.λ=40°.
A proton in a synchrotron is moving in a circle of radius 1 km and increasing its speed by v(t)=c1+c2t2,wherec1=2.0×105m/s,v(t)=c1+c2t2,wherec1=2.0×105m/s,
c2=105m/s3.c2=105m/s3. (a) What is the proton’s total acceleration at t = 5.0 s? (b) At what time does the expression for the velocity become unphysical?
A propeller blade at rest starts to rotate from t = 0 s to t = 5.0 s with a tangential acceleration of the tip of the blade at 3.00m/s2.3.00m/s2. The tip of the blade is 1.5 m from the axis of rotation. At t = 5.0 s, what is the total acceleration of the tip of the blade?
A particle is executing circular motion with a constant angular frequency of ω=4.00rad/s.ω=4.00rad/s. If time t = 0 corresponds to the position of the particle being located at y = 0 m and x = 5 m, (a) what is the position of the particle at t = 10 s? (b) What is its velocity at this time? (c) What is its acceleration?
A particle’s centripetal acceleration is aC=4.0m/s2aC=4.0m/s2 at t = 0 s. It is executing uniform circular motion about an axis at a distance of 5.0 m. What is its velocity at t = 10 s?
A rod 3.0 m in length is rotating at 2.0 rev/s about an axis at one end. Compare the centripetal accelerations at radii of (a) 1.0 m, (b) 2.0 m, and (c) 3.0 m.
A particle located initially at (1.5jˆ+4.0kˆ)m(1.5j^+4.0k^)m undergoes a displacement of (2.5iˆ+3.2jˆ−1.2kˆ)m.(2.5i^+3.2j^−1.2k^)m. What is the final position of the particle?
The position of a particle is given by r⃗ (t)=(50m/s)tiˆ−(4.9m/s2)t2jˆ.r→(t)=(50m/s)ti^−(4.9m/s2)t2j^. (a) What are the particle’s velocity and acceleration as functions of time? (b) What are the initial conditions to produce the motion?
A spaceship is traveling at a constant velocity of v⃗ (t)=250.0iˆm/sv→(t)=250.0i^m/s when its rockets fire, giving it an acceleration of a⃗ (t)=(3.0iˆ+4.0kˆ)m/s2.a→(t)=(3.0i^+4.0k^)m/s2. What is its velocity 55 s after the rockets fire?
A crossbow is aimed horizontally at a target 40 m away. The arrow hits 30 cm below the spot at which it was aimed. What is the initial velocity of the arrow?
A long jumper can jump a distance of 8.0 m when he takes off at an angle of 45°45° with respect to the horizontal. Assuming he can jump with the same initial speed at all angles, how much distance does he lose by taking off at 30°?30°?
On planet Arcon, the maximum horizontal range of a projectile launched at 10 m/s is 20 m. What is the acceleration of gravity on this planet?
A mountain biker encounters a jump on a race course that sends him into the air at 60°60° to the horizontal. If he lands at a horizontal distance of 45.0 m and 20 m below his launch point, what is his initial speed?
Which has the greater centripetal acceleration, a car with a speed of 15.0 m/s along a circular track of radius 100.0 m or a car with a speed of 12.0 m/s along a circular track of radius 75.0 m?
A geosynchronous satellite orbits Earth at a distance of 42,250.0 km and has a period of 1 day. What is the centripetal acceleration of the satellite?
Two speedboats are traveling at the same speed relative to the water in opposite directions in a moving river. An observer on the riverbank sees the boats moving at 4.0 m/s and 5.0 m/s. (a) What is the speed of the boats relative to the river? (b) How fast is the river moving relative to the shore?
Challenge Problems
World’s Longest Par 3. The tee of the world’s longest par 3 sits atop South Africa’s Hanglip Mountain at 400.0 m above the green and can only be reached by helicopter. The horizontal distance to the green is 359.0 m. Neglect air resistance and answer the following questions. (a) If a golfer launches a shot that is 40°40° with respect to the horizontal, what initial velocity must she give the ball? (b) What is the time to reach the green?
When a field goal kicker kicks a football as hard as he can at 45°45° to the horizontal, the ball just clears the 3-m-high crossbar of the goalposts 45.7 m away. (a) What is the maximum speed the kicker can impart to the football? (b) In addition to clearing the crossbar, the football must be high enough in the air early during its flight to clear the reach of the onrushing defensive lineman. If the lineman is 4.6 m away and has a vertical reach of 2.5 m, can he block the 45.7-m field goal attempt? (c) What if the lineman is 1.0 m away?
A truck is traveling east at 80 km/h. At an intersection 32 km ahead, a car is traveling north at 50 km/h. (a) How long after this moment will the vehicles be closest to each other? (b) How far apart will they be at that point?
