**AP Physics – Kinematics Example Problems**

1. An inclined railway travels at a speed of 1.10 m/s while climbing at an angle of 35.0°. The railway delivers riders to the top of a mountain – a change in elevation of 125 m. (a) Determine the rate at which the train’s elevation increases. (b) Determine the time for it to reach the top.

2. An airplane flying over Kansas with speed 410 km/h descends from an elevation of 4500 m to 2500 m in 145 seconds. (a) Determine the descent rate of the airplane. (b) Determine how far the plane has traveled over the ground during its descent.

3. A policeman’s radar is pointed in a northward direction and measures a car passing by that is traveling 10.0° east of north. The radar gun indicates 75 mph. What is the actual speed of the car? Hint: the radar gun only measures a component of the car’s velocity.

4. The pilot of an airplane notes an airspeed of 375 km/h and a heading of 30.0° while encountering a wind of 45.0 km/h blowing southward. Find the groundspeed and course.

5. A boy that can swim at 1.5 m/s heads directly across a river that flows at 0.80 m/s and is 20.0 m wide. (a) What is the speed of the boy relative to earth? (b) How much time does it take to cross? (c) How far downstream is he relative to his starting point on the other side? (d) Is there anyway he can cross in less time if he is not concerned where he lands? Explain.

6. In order to maintain a westward course and speed over ground of 20.0 km/h, what must be the speed through water and heading of a ship that encounters a current of 4.0 km/h southward?

7. Suppose the wind is 55.0 km/h, 30.0°. (a) In order to maintain a course of 90.0° in such a wind, what heading would be required of an airplane with airspeed 295 km/h? (b) If the destination is 1750 km, 90.0° how much time is “saved” due to the wind?

8. A
pilot flies from City A to City B. City B is a distance *d* due east from
City A. Suppose the plane has airspeed *v*_{PA} and encounters a
northerly wind with speed *v*_{AE}. (a) Solve for the flight
time, *t*, and the required heading of the airplane, θ, in terms of
other relevant variables. (b) Repeat this problem but now allow for a wind
with an arbitrary direction, φ.

9. A truck is traveling east at 50.0 mph. At an intersection 20.0 miles ahead, a car is moving north at 30.0 mph. How long after this moment will the two vehicles be closest to each other? How far apart will they be at that point?

10. A physics nerd is driving a car with a broken speedometer through falling snow. Relative to the earth, the snow falls straight down with speed 4.0 m/s. The nerd notes that the snow appears to be falling at an angle of 60.0° with respect to the vertical while he is driving. The nerd uses these facts to determine his speed – what is the result?

11. A ball is thrown with initial velocity 10.0 m/s, 0° out a window that is 15.0 m above the ground. (a) Determine the range of the ball. (b) Find the impact velocity.

12. A projectile is launched
horizontally from height *h* above a level surface. The object travels a
horizontal distance *x* before hitting. Derive equations for the initial
and final speed of the object in terms of *x*, *h*, and *g*.

13. A projectile is launched
over level ground with initial speed *v*_{o} and angle of
elevation *q*. Ignoring air
resistance, show that the projectile moves in a parabolic path. Derive
equations for range and for maximum height in terms of *v*_{o} and
*q* and show that maximum range is
achieved at *q* = 45°.

14. A kid shoots a basketball at an angle of 50.0° above the horizontal and makes a basket. The hoop that the ball goes through is located 4.0 m horizontally and 1.5 m vertically away from the point where the ball was released. Determine the velocity of the ball as it goes through the hoop.

15. Determine an equation for
making a free throw! The rim of the basket is 10 feet high and the foul line
is 13 feet 9 inches from the centerline of the rim. Assume that the ball is
launched from a height of 6 feet. Find the equation that defines all
successful free throws in terms of *v*_{o} and *θ*.
(Note: *g* = 32.2 ft/s^{2})

16. The position of a particle
is given by *x*(*t*) = 4.0 *t*^{2} and *y*(*t*)
= 0.5 *t*^{3} m. (a) What are the velocity and acceleration of
the particle at *t* = 2.0 s? (b) Determine the equation of the path that
the particle travels along.

17. At the same time a truck is
accelerating forward 3.0 m/s^{2} from rest, a pebble at the front end
of the truck’s bed is observed to accelerate backward 1.4 m across the bed in
1.5 s before falling out. The bed of the truck is 0.80 m above the road. (a)
Relative to the *truck*, what is the velocity of the pebble as it falls off?
(b) Relative to the *earth*, what is the velocity as it falls off? (c)
Relative to *earth*, what is the total horizontal distance traveled by the
pebble before it hits the ground? (d) Relative to the *truck*, what is
the total horizontal distance traveled before hitting? (e) Sketch the path of
the pebble’s motion as seen from each frame of reference.

18. A truck with initial
velocity 5.0 m/s north accelerates uniformly at 1.0 m/s^{2} north. At
the same instant a mouse in the bed of the truck accelerates from rest
uniformly at 3.0 m/s^{2} east, relative to the surface of the truck’s
bed. (a) Write the parametric equations *x*(*t*) and *y*(*t*)
that describe the mouse’s motion. (b) Relative to the earth, determine the
velocity and acceleration of the mouse when it has moved 1.2 m across the bed
of the truck from its original position. (b) Determine an equation that would
describe the path of the mouse’s motion relative to the earth.

19. An object moves along the
x-axis according to the function *x*(*t*) = 5 – 2*t* + 3*t*^{2}
– 0.5*t*^{3}, where *t* is time measured in seconds and x is
position measured in meters. (a) Determine expressions for this object’s
velocity and acceleration as functions of time. (b) For the interval of time *t*
= 0 to *t* = 4.0 s determine: (b) average acceleration, (c) maximum rate
of acceleration, (d) average velocity, (e) maximum speed, and (f) maximum
distance from the origin.

20. For each of the following
expressions determine the indefinite integral with respect to time:

(a) *v*(*t*) = 6*t*^{3} − 5*t*

(b) *a*(*t*) = 3*t*^{2} − 4*t* + 7

(c) *a*(*t*) = 10 + *t*^{−2}

(d) *v*(*t*) = 2*t*^{5} − *t*^{4}/3

(e) *v*(*t*) = 12*t*^{0.5} + 15

21. Beginning at a position 1.0
m to the right of the origin, an object moves along the *x*-axis with
velocity in m/s given by *v*(*t*) = 0.20*t*^{4} – *t*^{2},
where *t* is in seconds. (a) Determine its position, velocity, and
acceleration at *t* = 2.0 s. (b) For *t *> 0, determine the
greatest leftward position, speed, and acceleration.

22. A certain car is reported by a magazine to complete a “standing quarter mile” in 14.5 seconds with an ending speed of 95 mph or 139 ft/s. (a) Determine the average acceleration. (b) Assuming the acceleration is constant calculate the distance the car should travel in 14.5 seconds. How does this compare with one quarter mile (1320 ft)? Explain the discrepancy!

23. The magazine reported data
for time and speed of the same car as shown in the table below. Assuming that
speed is proportional to the square root of the time, find an equation for *v*(*t*)
to model this data. (a) Use this model to determine the functions *x*(*t*)
and *a*(*t*). (b) Calculate *x*(14.5 s) and compare to one
quarter mile. (c) The magazine reported the top speed of this car as 151 mph
or 221 ft/s. Determine the acceleration of the car as it reaches 151 mph
according to your functions. What does this result tell us about the
functions?

Time (s) |
Speed (mph) |
Converted Speed (ft/s) |

0 |
0 |
0 |

1.9 |
30 |
44.0 |

3.0 |
40 |
58.7 |

4.3 |
50 |
73.3 |

5.9 |
60 |
88.0 |

7.7 |
70 |
102.7 |

10.2 |
80 |
117.3 |

13.0 |
90 |
132.0 |

16.2 |
100 |
146.7 |

20.2 |
110 |
161.3 |

26.9 |
120 |
176.0 |

36.2 |
130 |
190.7 |

24. Try a different approach to
the car in the previous example: Assume that the acceleration decreases
linearly such that *a*(*t*) = *mt* + *b*. (a) Determine
values for *m* and *b* based on the known quarter mile (1320 ft) time
of 14.5 s and final speed of 139 ft/s. (b) What is the initial acceleration?
(c) Use this new model to determine the top speed of the car.

1. a.
0.631 m/s

b. 198 s

2. a.
13.8 m/s

b. 16.4 km

3. 76 mph

4. 355 km/h, 23.7°

5. a.
1.7 m/s

b. 13 s

c. 11 m

d. No

6. 20.4 km/h, 168.7°

7. a.
99.3°

b. 26.4 min

8. a. _{}

b. _{}

9. 0.29 hours later; 10.3 miles

10. 6.9 m/s

11. a. 17.5 m

b. 19.8 m/s, 300.3°

12. _{}

13. _{}

14. **v** = 5.36 m/s, 336.2°

15. _{} , where *θ* =
launch angle and *v*_{o} = initial speed (ft/s)

16. a.** v** = 17 m/s, 21°; **a**
= 1__0__ m/s^{2}, 37°

b. *y* = *x*^{1.5}/16

17. a. 1.87 m/s, W

b. 2.63 m/s, E

c. 3.04 m, E

d. 2.40 m, W

e.

18. a.* x*(*t*) = 1.5 *t*^{2};
*y*(*t*) = 5 *t* + 0.5 *t*^{2}

b. **v** = 6.5 m/s, 66°; **a** = 3.2 m/s^{2}, 18°

c. *y* = 4.08 *x*^{0.5} + 0.333 *x*

19. a. *v*(*t*) =
−2 + 6*t* − 1.5 *t*^{2}

*a*(*t*) = 6 − 3*t*

b. *a*_{avg} = 0

c. *a*_{max} = 6 m/s^{2}

d. **v**_{avg} = 2.00 m/s, right

e. *v*_{max} = 4.00 m/s

f. *x*_{max} = 13.4 m

20. a. *x*(*t*) = 1.5*t*^{4}
− 2.5*t*^{2} + *C*

b. *v*(*t*) = *t*^{3} − 2*t*^{2} + 7*t*
+ *C*

c. *v*(*t*) = 10*t* − *t*^{−1} + *C*

d. *x*(*t*) = *t*^{6}/3 − *t*^{5}/15
+ *C*

e. *x*(*t*) = 8*t*^{1.5} − 15*t* + *C*

21. a. **x** = 0.387 m, left
of origin; **v** = 0.800 m/s, left; **a** = 2.40 m/s^{2}, right

b. **x** = 0.4907 m, left; *v* = 1.25 m/s; **a** = 1.217 m/s^{2},
left

22. a. 9.61 ft/s^{2}

b. 1010 ft

23. *v* = 35.06 *t*^{
0.5}

a. *a* = 17.53 *t*^{ −0.5} ; *d* = 23.37 *t*^{
1.5}

b. 1290 ft (1/4 mile = 1320 ft)

c. *a* = 2.78 ft/s^{2} (should be zero!)

24. a. *m* = −1.229
ft/s^{3}; *b* = 18.497 ft/s^{2}

b. 18.5 ft/s^{2}

c. 139.2 ft/s