**AP Physics –
Dynamics Example Problems**

1. A
satellite of mass 50.0 kg is pulled by 45__0__ N of gravity. Small
thrusters are used to maneuver the satellite in its orbit. (a) What thrust
would cause the satellite to move with a constant velocity? Find the acceleration
of the satellite in response to the following thrusts: (b) 205 N up, (c) 205 N
down, and (d) 205 N right.

2. An
elevator car of mass 1570 kg is raised and lowered by a cable. Determine the
force that the cable exerts on the car under the following circumstances: (a)
car accelerates upward at 2.00 m/s^{2} from rest, (b) car rises at a
constant 3.00 m/s, (c) car decelerates 1.50 m/s^{2} while rising, and
(d) car lowers at a constant 3.00 m/s.

3. A balloon of a known mass or weight is dropped from a known height and timed. Determine the average amount of air resistance that acts on it.

4. An
object that weighs 20.0 N is thrown by a person. Determine the resulting
acceleration if the force exerted by the person is: (a) 30.0 N downward, and
(b) 30.0 N rightward. (c) What force would the person have to exert in order
that the object accelerate 4.00 m/s^{2} directly to the left?

5. A certain rope has a tensile strength of 5.0 kN – this is the greatest force it can exert without breaking. Suppose this rope is used for rappelling. What is the maximum deceleration of a 50.0 kg rappeller that the rope can withstand? Repeat for a 100.0 kg person.

6. An
object of mass 5.0 kg is subjected to a rightward force in newtons of *F*
= 3 *t*^{2} – 4 *t* where *t* is measured in seconds.
The object has velocity **v** = 7 m/s, left at *t* = 0. Determine the
acceleration and velocity of the object at *t* = 9.0 s.

7. As
shown in the diagram below a mass *m* is pulled by two forces but remains
at rest due to a third force – gravity. (a) Determine the value of each force
in terms of relevant variables and constants. (b) If force *F*_{2}
suddenly ceases to exist what will be the instantaneous acceleration of the
mass?

8. Two
astronauts aboard the space station bump into one another. Astronaut Jones,
mass 90.0 kg is floating to the right and collides with Astronaut Smith, mass
80.0 kg initially at rest. If Smith is accelerated 2.00 m/s^{2}, right
find: (a) The resulting acceleration of Jones, (b) the force that Jones exerts
on Smith, and (c) the force that Smith exerts on Jones.

9. A doomed skydiver of mass 85.0 kg whose chute failed to open tries to slow himself by removing his 10.0 kg pack and throwing it downward. He exerts a downward force of 50.0 N on the pack. (a) Ignoring air resistance, determine the resulting accelerations of the skydiver and the pack. (b) What if air resistance is not ignored and the skydiver is at terminal velocity?

10. Use a CBR to measure a book of known weight that falls and is caught by a student. Use the velocity vs. time graph and your knowledge of physics to find the force that the falling book exerts on the student’s hands.

11. A student of mass 75.0 kg
stands at rest on the floor. (a) Find the normal force the floor exerts on his
feet. (b) Find the normal force of his feet on the floor if he jumps upward
with acceleration 2.50 m/s^{2}. (c) Find the normal force if he pushes
downward on a nearby table with a force of 50.0 N.

12. A bowling ball is prevented from rolling down an incline of 30.0° by a student pulling parallel to the incline. Based on the known weight of the ball determine the normal force that acts upon it and the amount of force that the student has to exert.

13. A 500.0 gram mass sits atop a 1.40 kg book. A force of 30.0 N upward is applied to the bottom of the book. (a) Analyze as one object to determine the acceleration. (b) Analyze the forces on the mass to solve for the force that the book exerts on it. (c) Analyze the forces on the book to solve for the force that the mass exerts on it.

14. A man of mass 100.0 kg
climbs a rope of mass 2.00 kg that hangs from the ceiling of a gym. Every time
the man lifts himself a bit higher he accelerates upward 0.50 m/s^{2}.
Determine the maximum tension in the rope that occurs during his climb to the
top. At what part of the rope does this occur? Where is the man when it
occurs?

15. Determine expressions for the acceleration and tension for Atwood’s machine, assuming a massless, frictionless pulley and massless string.

16. An object of mass *M*
is pulled across a level surface without friction by a string that passes over
a pulley. At the end of the string is an object with mass *m*. Determine
the acceleration and the tension in the string, assuming massless string and a
massless, frictionless pulley.

17. Examine the diagram below.
Solve for the acceleration of each mass. Ignore friction and mass of the
string and pulleys.

18. The coefficients of friction
for an old physics book upon a shelf are: μ_{s} = 0.50 and μ_{k}
= 0.40. Suppose the book has weight 14.5 N and a horizontal force is applied
to it. (a) Determine the maximum force that can be applied without moving the
book. (b) Determine the force required to keep the book moving at a constant
speed across the shelf. (c) If the magnitude of the applied force is increased
gradually, find the initial acceleration of the book just as it starts to move.

19. A car with speed *v*_{o}
applies the brakes and slows to a stop. Derive and simplify an equation for
the stopping distance *d* in terms of *v*_{o} and μ.

20. Measure the mass and sliding deceleration of a block across a level surface. Then add 500.0 g to the block and accelerate it with 400.0 g in a modified Atwood’s machine. (a) Determine the coefficient of sliding friction. (b) Find the acceleration when pulled by the 400.0 g mass. (c) Find the tension in the string as it is pulled.

21. Measure the mass and coasting deceleration of the fan cart. Measure the acceleration of the cart under the influence of the fan. (a) Determine μ. (b) Determine the thrust of the fan. (c) Predict the acceleration of the cart loaded with a 200.0 g mass moving in either direction under the influence of the fan.

22. A traveler pulls a suitcase
of mass 8.00 kg across a level surface by pulling on the handle 20.0 N at an
angle of 50.0° relative to horizontal. Friction against the suitcase can be
modeled by μ_{k} = 0.100. (a) Determine the acceleration of the
suitcase. (b) What amount of force applied at the same angle would be needed
to keep the suitcase moving at constant velocity?

23. A horse of mass 509 kg pulls
a sleigh of mass 255 kg and both horse and sleigh accelerate 0.500 m/s^{2}.
The coefficient of friction for the sleigh is 0.15 as it moves over the snow.
(a) Find the force that the horse must exert on the sleigh. (b) Determine the
amount of horizontal force that the horse’s feet must exert.

24. Consider the previous
problem generically. An animal or vehicle of mass *M* pulls an object of
mass *m* across a level surface. Derive an expression for the largest
value of *m* in terms of μ_{s} for the animal/vehicle, μ_{k}
for the pulled object, and the acceleration *a* of both.

25. A baseball has mass 0.145 kg
and has a terminal velocity of 45 m/s. Suppose that air resistance against the
ball is modeled with the formula **F*** _{drag} = *–

26. Repeat the above problem
using the model *F _{drag} = *

27. Another baseball is prepared
that has the same diameter and the same external surface as the first, but
which has a mass of 1.45 kg (it is more dense and has more mass inside). What
is the terminal velocity of this more massive ball based on the model *F _{drag}
= *

28. A ping-pong ball of mass 2.3
g is released from a state of rest. When it reaches a speed of 2.0 m/s it has
an acceleration of 7.6 m/s^{2} downward. Suppose air resistance is
modeled by **F*** _{drag} = *–

29. Fill a balloon with air and
measure its apparent mass. Use a ranging device to measure its motion when
tossed into the air and allowed to fall. Note the terminal velocity and the
acceleration when it reaches its maximum height. (a) Determine the mass of the
balloon (and air inside it). (b) Find the force of buoyancy. (c) Determine
the value of *k* in *D* = *kv*. (d) Use the initial upward
velocity of the balloon to determine *v*(*t*) – compare to an
automatic curve fit.

30. The 1997 Dodge Viper has
mass 1547 kg and an engine that can produce a maximum driving force of 12.36
kN. Suppose drag is proportional to speed such that *k* = 164 Ns/m.

(a) Determine the initial acceleration of the Viper from rest. (b) Determine
the maximum speed (i.e. terminal velocity). (c) Calculate the distance and
speed for *t* = 12.2 s. (d) Find time and distance for the car to reach
99% of its top speed. (e) Graph *v*(*t*) and the actual data below
taken from a car magazine by and compare the ¼ mile results to part (c).

Time (s) |
Speed (m/s) |

0.0 |
0.00 |

1.7 |
13.4 |

2.4 |
17.9 |

3.1 |
22.4 |

4.0 |
26.8 |

4.9 |
31.3 |

5.9 |
35.8 |

7.4 |
40.2 |

8.7 |
44.7 |

10.6 |
49.2 |

12.5 |
53.6 |

14.7 |
58.1 |

18.3 |
62.6 |

22.5 |
67.1 |

28.2 |
71.5 |

¼ mile data: 12.2 s, 52.8 m/s |
Top speed: 79.1 m/s |

31. A book is placed on a board tilted at an angle. Given the coefficients of static and kinetic friction and the value of the angle, solve for: (a) the force applied parallel to the surface necessary to prevent the book from moving, and (b) the acceleration of the book if it is released on the ramp.

32. A rollercoaster ride is
shown in the diagram below. The mass of the car and rider is 700.0 kg and the
coefficient of friction is 0.0500. (a) Determine the force that must be applied
to the car as it is pulled up the hill at a constant speed of 1.00 m/s. (b)
The car is released at point A with speed 1.00 m/s – find the speed at point B.

33. A car moves at constant
velocity on a level roadway but then encounters a hill that is inclined an
amount θ. Assuming the driver does nothing with the gas or brakes,
determine the deceleration that will occur in terms of *g*, μ, and
θ. Disregard velocity dependent forces.

34. A snow skier glides with constant velocity on a slope inclined at an angle of 5.0°. The skier then encounters a ski trail with an incline of 30.0°. Determine the rate of acceleration.

__Answers__

1. a.
45__0__ N, up

b. 4.90 m/s^{2}, down

c. 13.1 m/s^{2}, down

d. 9.89 m/s^{2}, 294.5°

2. a.
18.5 kN, up

b. 15.4 kN, up

c. 13.0 kN, up

d. 15.4 kN, up

3.

4. a.
24.5 m/s^{2}, down

b. 17.7 m/s^{2}, 326.3°

c. 21.6 N, 67.8°

5. 9__0__
m/s^{2}, 4__0__ m/s^{2}

6. 41.4
m/s^{2}, 106 m/s

7. a. _{}

b. _{} , left

8. a.
1.78 m/s^{2}, left

b. 16__0__ N, right

c. 16__0__ N, left

9. a.
9.21 m/s^{2}, down; 14.8 m/s^{2}, down

b. 0.588 m/s^{2}, up; 5.00 m/s^{2}, down

10.

11. a. 735 N, up

b. 923 N, down

c. 685 N, up (on feet)

12. *F*_{N} = 0.866*W*;
*F*_{A} = 0.500*W*

13. a. 5.99 m/s^{2}, up

b. 7.89 N, up

c. 7.89 N, down

14. 1050 N at top of rope any
time

the man accelerates upward

15. _{} (*m*_{1} down,
*m*_{2} up)

_{}

16. _{}

_{}

17. _{} , down

_{} , up

18. a. 7.3 N

b. 5.8 N

c. 0.98 m/s^{2}

19. _{}

20.

21.

22. a. 0.818 m/s^{2},
forward

b. 10.9 N

23. a. 458 N, forward

b. 713 N, backward

24. _{}

25. a. 0.0316 kg/s

b. 14.2 m/s^{2}, down

c. 3.27 m/s^{2}, down

d. 13.1 m/s^{2}, 228.4°

26. a. 7.02 × 10^{−4}
kg/m

b. 11.7 m/s^{2}, down

c. 5.44 m/s^{2}, down

d. 12.5 m/s^{2}, 231.7°

27. 142 m/s

28. a. 8.91 m/s

b. taking down to be positive:

_{}

_{}

_{}

c. 0.913 s, 5.65 m/s

29.

30. a. 7.99 m/s^{2}

b. 75.4 m/s

c. 404 m, 54.7 m/s

d. 43.4 s, 2570 m

e.

31.

_{}

32. a. 4220 N parallel to
incline

b. 26.0 m/s

33. _{}

34. 4.2 m/s^{2}