A Boat Crossing a River
S UMMARY
If a particle moves with constant acceleration a and has velocity v
iand position r
iat
t⫽0, its velocity and position vectors at some later time t are
(4.8) (4.9)
For two-dimensional motion in the xy plane under constant acceleration, each of these vector expressions is equivalent to two component expressions — one for the motion in the x direction and one for the motion in the y direction. You should be able to break the two-dimensional motion of any object into these two compo- nents.
Projectile motion is one type of two-dimensional motion under constant ac- celeration, where and It is useful to think of projectile motion as the superposition of two motions: (1) constant-velocity motion in the x direction and (2) free-fall motion in the vertical direction subject to a constant downward acceleration of magnitude g
⫽9.80 m/s
2. You should be able to analyze the mo- tion in terms of separate horizontal and vertical components of velocity, as shown in Figure 4.24.
A particle moving in a circle of radius r with constant speed v is in uniform circular motion. It undergoes a centripetal (or radial) acceleration a
rbecause the direction of v changes in time. The magnitude of a
ris
(4.18)
and its direction is always toward the center of the circle.
If a particle moves along a curved path in such a way that both the magnitude and the direction of v change in time, then the particle has an acceleration vector that can be described by two component vectors: (1) a radial component vector a
rthat causes the change in direction of v and (2) a tangential component vector a
tthat causes the change in magnitude of v. The magnitude of a
ris
v2/r, and the magnitude of a
tis You should be able to sketch motion diagrams for an object following a curved path and show how the velocity and acceleration vectors change as the object’s motion varies.
The velocity v of a particle measured in a fixed frame of reference S can be re- lated to the velocity vⴕ of the same particle measured in a moving frame of refer- ence S⬘ by
(4.21)
where v
0is the velocity of S⬘ relative to S. You should be able to translate back and forth between different frames of reference.
vⴕ
⫽v
⫺v
0d兩
v
兩/dt.
ar⫽ v2 r ay⫽ ⫺g.
ax⫽
0
r
f⫽r
i⫹v
it⫹12at
2v
f⫽v
i⫹at
Figure 4.24
Analyzing projectile motion in terms of horizontal and vertical components.Projectile motion is equivalent to…
vi
i
(x, y) y
x x
i
Horizontal motion at constant velocity
vyi
Vertical motion at constant acceleration θ
vxf = vxi = vi cos
θ
and…
y vyf
Q UESTIONS
and therefore has no acceleration. The professor claims that the student is wrong because the satellite must have a centripetal acceleration as it moves in its circular orbit.
What is wrong with the student’s argument?
12. What is the fundamental difference between the unit vec- tors and and the unit vectors iand j?
13. At the end of its arc, the velocity of a pendulum is zero. Is its acceleration also zero at this point?
14. If a rock is dropped from the top of a sailboat’s mast, will it hit the deck at the same point regardless of whether the boat is at rest or in motion at constant velocity?
15. A stone is thrown upward from the top of a building.
Does the stone’s displacement depend on the location of the origin of the coordinate system? Does the stone’s ve- locity depend on the location of the origin?
16. Is it possible for a vehicle to travel around a curve without accelerating? Explain.
17. A baseball is thrown with an initial velocity of (10i⫹15j) m/s. When it reaches the top of its trajectory, what are (a) its velocity and (b) its acceleration? Neglect the effect of air resistance.
18. An object moves in a circular path with constant speed v.
(a) Is the velocity of the object constant? (b) Is its acceler- ation constant? Explain.
19. A projectile is fired at some angle to the horizontal with some initial speed vi, and air resistance is neglected. Is the projectile a freely falling body? What is its accelera- tion in the vertical direction? What is its acceleration in the horizontal direction?
20. A projectile is fired at an angle of 30° from the horizontal with some initial speed. Firing at what other projectile an- gle results in the same range if the initial speed is the same in both cases? Neglect air resistance.
21. A projectile is fired on the Earth with some initial velocity.
Another projectile is fired on the Moon with the same ini- tial velocity. If air resistance is neglected, which projectile has the greater range? Which reaches the greater alti- tude? (Note that the free-fall acceleration on the Moon is about 1.6 m/s2.)
22. As a projectile moves through its parabolic trajectory, which of these quantities, if any, remain constant:
(a) speed, (b) acceleration, (c) horizontal component of velocity, (d) vertical component of velocity?
23. A passenger on a train that is moving with constant veloc- ity drops a spoon. What is the acceleration of the spoon relative to (a) the train and (b) the Earth?
ˆ r ˆ 1. Can an object accelerate if its speed is constant? Can an
object accelerate if its velocity is constant?
2. If the average velocity of a particle is zero in some time in- terval, what can you say about the displacement of the particle for that interval?
3. If you know the position vectors of a particle at two points along its path and also know the time it took to get from one point to the other, can you determine the particle’s instantaneous velocity? Its average velocity? Explain.
4. Describe a situation in which the velocity of a particle is always perpendicular to the position vector.
5. Explain whether or not the following particles have an ac- celeration: (a) a particle moving in a straight line with constant speed and (b) a particle moving around a curve with constant speed.
6. Correct the following statement: “The racing car rounds the turn at a constant velocity of 90 mi/h.’’
7. Determine which of the following moving objects have an approximately parabolic trajectory: (a) a ball thrown in an arbitrary direction, (b) a jet airplane, (c) a rocket leav- ing the launching pad, (d) a rocket whose engines fail a few minutes after launch, (e) a tossed stone moving to the bottom of a pond.
8. A rock is dropped at the same instant that a ball at the same initial elevation is thrown horizontally. Which will have the greater speed when it reaches ground level?
9. A spacecraft drifts through space at a constant velocity.
Suddenly, a gas leak in the side of the spacecraft causes a constant acceleration of the spacecraft in a direction per- pendicular to the initial velocity. The orientation of the spacecraft does not change, and so the acceleration re- mains perpendicular to the original direction of the ve- locity. What is the shape of the path followed by the spacecraft in this situation?
10. A ball is projected horizontally from the top of a building.
One second later another ball is projected horizontally from the same point with the same velocity. At what point in the motion will the balls be closest to each other? Will the first ball always be traveling faster than the second ball? How much time passes between the moment the first ball hits the ground and the moment the second one hits the ground? Can the horizontal projection velocity of the second ball be changed so that the balls arrive at the ground at the same time?
11. A student argues that as a satellite orbits the Earth in a circular path, the satellite moves with a constant velocity
P ROBLEMS
6. The vector position of a particle varies in time accord- ing to the expression r⫽(3.00i⫺6.00t2j) m. (a) Find expressions for the velocity and acceleration as func- tions of time. (b) Determine the particle’s position and velocity at t⫽1.00 s.
7. A fish swimming in a horizontal plane has velocity vi⫽(4.00i⫹1.00j) m/s at a point in the ocean whose displacement from a certain rock is ri⫽(10.0i⫺4.00j) m. After the fish swims with constant acceleration for 20.0 s, its velocity is v⫽(20.0i⫺5.00j) m/s. (a) What are the components of the acceleration? (b) What is the direction of the acceleration with respect to the unit vec- tor i? (c) Where is the fish at t⫽25.0 s if it maintains its original acceleration and in what direction is it moving?
8. A particle initially located at the origin has an accelera- tion of a⫽3.00jm/s2and an initial velocity of vi⫽ 5.00im/s. Find (a) the vector position and velocity at any time tand (b) the coordinates and speed of the particle at t⫽2.00 s.
Section 4.3 Projectile Motion
(Neglect air resistance in all problems and take g⫽ 9.80 m/s2.)
9. In a local bar, a customer slides an empty beer mug down the counter for a refill. The bartender is momen- tarily distracted and does not see the mug, which slides off the counter and strikes the floor 1.40 m from the base of the counter. If the height of the counter is 0.860 m, (a) with what velocity did the mug leave the counter and (b) what was the direction of the mug’s velocity just before it hit the floor?
10. In a local bar, a customer slides an empty beer mug down the counter for a refill. The bartender is momen- tarily distracted and does not see the mug, which slides off the counter and strikes the floor at distance dfrom the base of the counter. If the height of the counter is h, (a) with what velocity did the mug leave the counter and (b) what was the direction of the mug’s velocity just before it hit the floor?
11. One strategy in a snowball fight is to throw a first snow- ball at a high angle over level ground. While your oppo- nent is watching the first one, you throw a second one at a low angle and timed to arrive at your opponent be- fore or at the same time as the first one. Assume both snowballs are thrown with a speed of 25.0 m/s. The first one is thrown at an angle of 70.0° with respect to the horizontal. (a) At what angle should the second (low- angle) snowball be thrown if it is to land at the same point as the first? (b) How many seconds later should Section 4.1 The Displacement, Velocity, and Acceleration
Vectors
1. A motorist drives south at 20.0 m/s for 3.00 min, then turns west and travels at 25.0 m/s for 2.00 min, and fi- nally travels northwest at 30.0 m/s for 1.00 min. For this 6.00-min trip, find (a) the total vector displacement, (b) the average speed, and (c) the average velocity. Use a coordinate system in which east is the positive xaxis.
2. Suppose that the position vector for a particle is given
as with and where
m/s, m, m/s2, and
m. (a) Calculate the average velocity during the time in- terval from s to s. (b) Determine the velocity and the speed at s.
3. A golf ball is hit off a tee at the edge of a cliff. Its xand y coordinates versus time are given by the following ex- pressions:
and
(a) Write a vector expression for the ball’s position as a function of time, using the unit vectors iand j. By taking derivatives of your results, write expressions for (b) the velocity vector as a function of time and (c) the accelera- tion vector as a function of time. Now use unit vector no- tation to write expressions for (d) the position, (e) the velocity, and (f) the acceleration of the ball, all at t⫽3.00 s.
4. The coordinates of an object moving in the xyplane vary with time according to the equations
and
where tis in seconds and has units of seconds⫺1. (a) Determine the components of velocity and compo- nents of acceleration at t⫽0. (b) Write expressions for the position vector, the velocity vector, and the accelera- tion vector at any time (c) Describe the path of the object on an xygraph.
Section 4.2 Two-Dimensional Motion with Constant Acceleration
5. At t⫽0, a particle moving in the xyplane with constant acceleration has a velocity of
when it is at the origin. At t⫽3.00 s, the particle’s ve- locity is v⫽(9.00i⫹7.00j) m/s. Find (a) the accelera- tion of the particle and (b) its coordinates at any time t.
vi⫽(3.00i⫺2.00j) m/s t⬎0.
y⫽(4.00 m)⫺(5.00 m)cos t x⫽ ⫺(5.00 m) sin t y⫽(4.00 m/s)t⫺(4.90 m/s2)t2
x⫽(18.0 m/s)t t⫽2.00 t⫽4.00 t⫽2.00
d⫽1.00 c⫽0.125
b⫽1.00 a⫽1.00
y⫽ct2⫹d, x⫽at⫹b
r⫽xi⫹yj,
1, 2, 3= straightforward, intermediate, challenging = full solution available in the Student Solutions Manual and Study Guide
WEB= solution posted at http://www.saunderscollege.com/physics/ = Computer useful in solving problem = Interactive Physics
= paired numerical/symbolic problems
WEB WEB
the second snowball be thrown if it is to land at the same time as the first?
12. A tennis player standing 12.6 m from the net hits the ball at 3.00° above the horizontal. To clear the net, the ball must rise at least 0.330 m. If the ball just clears the net at the apex of its trajectory, how fast was the ball moving when it left the racket?
13. An artillery shell is fired with an initial velocity of 300 m/s at 55.0° above the horizontal. It explodes on a mountainside 42.0 s after firing. What are the xand y coordinates of the shell where it explodes, relative to its firing point?
14. An astronaut on a strange planet finds that she can jump a maximum horizontal distance of 15.0 m if her initial speed is 3.00 m/s. What is the free-fall accelera- tion on the planet?
15. A projectile is fired in such a way that its horizontal range is equal to three times its maximum height. What is the angle of projection? Give your answer to three sig- nificant figures.
16. A ball is tossed from an upper-story window of a build- ing. The ball is given an initial velocity of 8.00 m/s at an angle of 20.0° below the horizontal. It strikes the ground 3.00 s later. (a) How far horizontally from the base of the building does the ball strike the ground?
(b) Find the height from which the ball was thrown.
(c) How long does it take the ball to reach a point 10.0 m below the level of launching?
17. A cannon with a muzzle speed of 1 000 m/s is used to start an avalanche on a mountain slope. The target is 2 000 m from the cannon horizontally and 800 m above the cannon. At what angle, above the horizontal, should the cannon be fired?
18. Consider a projectile that is launched from the origin of an xycoordinate system with speed viat initial angle i
above the horizontal. Note that at the apex of its trajec- tory the projectile is moving horizontally, so that the slope of its path is zero. Use the expression for the tra- jectory given in Equation 4.12 to find the xcoordinate that corresponds to the maximum height. Use this xco- ordinate and the symmetry of the trajectory to deter- mine the horizontal range of the projectile.
19. A placekicker must kick a football from a point 36.0 m (about 40 yards) from the goal, and half the crowd hopes the ball will clear the crossbar, which is 3.05 m high. When kicked, the ball leaves the ground with a speed of 20.0 m/s at an angle of 53.0° to the horizontal.
(a) By how much does the ball clear or fall short of clearing the crossbar? (b) Does the ball approach the crossbar while still rising or while falling?
20. A firefighter 50.0 m away from a burning building di- rects a stream of water from a fire hose at an angle of 30.0° above the horizontal, as in Figure P4.20. If the speed of the stream is 40.0 m/s, at what height will the water strike the building?
21. A firefighter a distance dfrom a burning building di- rects a stream of water from a fire hose at angle iabove the horizontal as in Figure P4.20. If the initial speed of the stream is vi, at what height hdoes the water strike the building?
22. A soccer player kicks a rock horizontally off a cliff 40.0 m high into a pool of water. If the player hears the sound of the splash 3.00 s later, what was the initial speed given to the rock? Assume the speed of sound in air to be 343 m/s.
vi
d
h
θi
Figure P4.20
Problems 20 and 21.(Frederick McKinney/FPG Interna- tional)WEB
23. A basketball star covers 2.80 m horizontally in a jump to dunk the ball (Fig. P4.23). His motion through space can be modeled as that of a particle at a point called his center of mass (which we shall define in Chapter 9). His center of mass is at elevation 1.02 m when he leaves the floor. It reaches a maximum height of 1.85 m above the floor and is at elevation 0.900 m when he touches down again. Determine (a) his time of flight (his “hang time”), (b) his horizontal and (c) vertical velocity com- ponents at the instant of takeoff, and (d) his takeoff an- gle. (e) For comparison, determine the hang time of a whitetail deer making a jump with center-of-mass eleva- tions yi⫽1.20m, ymax⫽2.50m, yf⫽0.700m.
Section 4.4 Uniform Circular Motion
24. The orbit of the Moon about the Earth is approximately circular, with a mean radius of 3.84⫻108m. It takes 27.3 days for the Moon to complete one revolution about the Earth. Find (a) the mean orbital speed of the Moon and (b) its centripetal acceleration.
25. The athlete shown in Figure P4.25 rotates a 1.00-kg dis- cus along a circular path of radius 1.06 m. The maximum speed of the discus is 20.0 m/s. Determine the magni- tude of the maximum radial acceleration of the discus.
WEB
Figure P4.23
(Top, Ron Chapple/FPG International;bottom, Bill Lea/Dembinsky Photo Associates)
Figure P4.25
(Sam Sargent/Liaison International)26. From information on the endsheets of this book, com- pute, for a point located on the surface of the Earth at the equator, the radial acceleration due to the rotation of the Earth about its axis.
27. A tire 0.500 m in radius rotates at a constant rate of 200 rev/min. Find the speed and acceleration of a small stone lodged in the tread of the tire (on its outer edge).
(Hint:In one revolution, the stone travels a distance equal to the circumference of its path, 2r.)
28. During liftoff, Space Shuttle astronauts typically feel ac- celerations up to 1.4g, where g⫽9.80 m/s2. In their training, astronauts ride in a device where they experi- ence such an acceleration as a centripetal acceleration.
Specifically, the astronaut is fastened securely at the end of a mechanical arm that then turns at constant speed in a horizontal circle. Determine the rotation rate, in revolutions per second, required to give an astronaut a centripetal acceleration of 1.40g while the astronaut moves in a circle of radius 10.0 m.
29. Young David who slew Goliath experimented with slings before tackling the giant. He found that he could re- volve a sling of length 0.600 m at the rate of 8.00 rev/s.
If he increased the length to 0.900 m, he could revolve the sling only 6.00 times per second. (a) Which rate of rotation gives the greater speed for the stone at the end of the sling? (b) What is the centripetal acceleration of the stone at 8.00 rev/s? (c) What is the centripetal ac- celeration at 6.00 rev/s?
30. The astronaut orbiting the Earth in Figure P4.30 is preparing to dock with a Westar VI satellite. The satel- lite is in a circular orbit 600 km above the Earth’s sur- face, where the free-fall acceleration is 8.21 m/s2. The radius of the Earth is 6 400 km. Determine the speed of the satellite and the time required to complete one or- bit around the Earth.
at a given instant of time. At this instant, find (a) the ra- dial acceleration, (b) the speed of the particle, and (c) its tangential acceleration.
34. A student attaches a ball to the end of a string 0.600 m in length and then swings the ball in a vertical circle.
The speed of the ball is 4.30 m/s at its highest point and 6.50 m/s at its lowest point. Find the acceleration of the ball when the string is vertical and the ball is at (a) its highest point and (b) its lowest point.
35. A ball swings in a vertical circle at the end of a rope 1.50 m long. When the ball is 36.9° past the lowest point and on its way up, its total acceleration is (⫺22.5i⫹ 20.2j) m/s2. At that instant, (a) sketch a vector diagram showing the components of this acceleration, (b) deter- mine the magnitude of its radial acceleration, and (c) determine the speed and velocity of the ball.
Section 4.6 Relative Velocity and Relative Acceleration 36. Heather in her Corvette accelerates at the rate of
(3.00i⫺2.00j) m/s2, while Jill in her Jaguar accelerates at (1.00i⫹3.00j) m/s2. They both start from rest at the origin of an xycoordinate system. After 5.00 s, (a) what is Heather’s speed with respect to Jill, (b) how far apart are they, and (c) what is Heather’s acceleration relative to Jill?
37. A river has a steady speed of 0.500 m/s. A student swims upstream a distance of 1.00 km and swims back to the starting point. If the student can swim at a speed of 1.20 m/s in still water, how long does the trip take?
Compare this with the time the trip would take if the water were still.
38. How long does it take an automobile traveling in the left lane at 60.0 km/h to pull alongside a car traveling in the right lane at 40.0 km/h if the cars’ front bumpers are initially 100 m apart?
39. The pilot of an airplane notes that the compass indi- cates a heading due west. The airplane’s speed relative to the air is 150 km/h. If there is a wind of 30.0 km/h toward the north, find the velocity of the airplane rela- tive to the ground.
40. Two swimmers, Alan and Beth, start at the same point in a stream that flows with a speed v. Both move at the same speed c(c⬎v) relative to the stream. Alan swims downstream a distance Land then upstream the same distance. Beth swims such that her motion relative to the ground is perpendicular to the banks of the stream.
She swims a distance Lin this direction and then back.
The result of the motions of Alan and Beth is that they both return to the starting point. Which swimmer re- turns first? (Note:First guess at the answer.)
41. A child in danger of drowning in a river is being carried downstream by a current that has a speed of 2.50 km/h.
The child is 0.600 km from shore and 0.800 km up- stream of a boat landing when a rescue boat sets out.
(a) If the boat proceeds at its maximum speed of 20.0 km/h relative to the water, what heading relative to the shore should the pilot take? (b) What angle does
Figure P4.30
(Courtesy of NASA)Figure P4.33
30.0°
2.50 m a
v a = 15.0 m/s2
Section 4.5 Tangential and Radial Acceleration 31. A train slows down as it rounds a sharp horizontal
curve, slowing from 90.0 km/h to 50.0 km/h in the 15.0 s that it takes to round the curve. The radius of the curve is 150 m. Compute the acceleration at the mo- ment the train speed reaches 50.0 km/h. Assume that the train slows down at a uniform rate during the 15.0-s interval.
32. An automobile whose speed is increasing at a rate of 0.600 m/s2travels along a circular road of radius 20.0 m.
When the instantaneous speed of the automobile is 4.00 m/s, find (a) the tangential acceleration component, (b) the radial acceleration component, and (c) the magnitude and direction of the total acceleration.
33. Figure P4.33 shows the total acceleration and velocity of a particle moving clockwise in a circle of radius 2.50 m