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AP Physics C: Mechanics Full-length practice exam 1

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Full-length practice exam 1

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Question 1

A 0.1 kg bullet traveling at $400$ m/s embeds in a 4.9 kg stationary block on a frictionless surface. A spring ( $k = 1000$ N/m) is in front of the block. What is the maximum spring compression?

0.2

mB.

0.4

mC.

0.8

mD.

0.566

Question 2

A car rounds a banked curve of radius $R$ at angle $θ$ (no friction). What is the speed $v$ at which no friction is needed?

R g sin θ

R g cos θ

R g tan θ

R g tan θ

Question 3

Three particles have masses and positions: $m_{1} = 2$ kg at $x_{1} = 0$ m, $m_{2} = 3$ kg at $x_{2} = 4$ m, and $m_{3} = 1$ kg at $x_{3} = 6$ m. Find the center of mass.

\frac{10}{3} \approx 3.33

mB.

3

mC.

2

mD.

4

Question 4

A projectile is launched at angle $θ$ above the horizontal with initial speed $v_{0}$ . Which expression gives the maximum height?

\frac{v _{0}^{2}}{2 g}

\frac{v _{0}^{2} sin θ}{2 g}

\frac{v _{0}^{2} sin ^{2} θ}{2 g}

\frac{v _{0}^{2} cos ^{2} θ}{2 g}

Question 5

A 5 kg block is released from rest at the top of a frictionless ramp of height $h = 3$ m and then slides along a rough horizontal surface ( $μ_{k} = 0.2$ , $g = 10$ m/s²). How far does it travel on the rough surface before stopping?

3

mB.

7.5

mC.

15

mD.

30

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Question 6

A figure skater with $I_{1} = 4$ kg·m² spins at $2$ rad/s and pulls in arms to reach $I_{2} = 1$ kg·m². What is the new angular velocity?

2

rad/sB.

4

rad/sC.

8

rad/sD.

0.5

rad/s

Question 7

In a one-dimensional elastic collision between mass $m_{1}$ (initial speed $v_{0}$ ) and stationary mass $m_{2}$ , the final velocities are $v_{1}^{'} = \frac{m _{1} - m _{2}}{m _{1} + m _{2}} v_{0}$ and $v_{2}^{'} = \frac{2 m _{1}}{m _{1} + m _{2}} v_{0}$ . Verify these satisfy energy conservation symbolically.

A.Only

v_{2}^{'}

conserves energy;

v_{1}^{'}

does not.B.Neither velocity satisfies energy conservation.C.Both velocities together conserve momentum but not energy.D.Both velocities together conserve both momentum and kinetic energy.

Question 8

A satellite is transferred from a circular orbit of radius $r_{1}$ to a circular orbit of radius $r_{2} > r_{1}$ via a Hohmann transfer (half-ellipse). During the transfer, which statement about speed is correct at the outer transfer point?

A.The satellite must decelerate to enter the ellipse and decelerate again to circularize at

r_{2}

.B.The satellite accelerates at

r_{1}

, coasts on the ellipse, and decelerates at

r_{2}

to reach the circular orbit speed.C.The satellite accelerates at

r_{1}

, coasts on the ellipse, then accelerates again at

r_{2}

to match the circular orbit speed.D.The satellite's speed at

r_{2}

on the ellipse equals the circular orbit speed at

r_{2}

, so no second burn is needed.

Question 9

A motor lifts a 200 kg load at a constant velocity of $2$ m/s. What power does the motor deliver? ( $g = 10$ m/s²)

400

WB.

1000

WC.

4000

WD.

2000

Question 10

A 3 kg object moving at $8$ m/s to the right undergoes a perfectly inelastic collision with a 5 kg object moving at $4$ m/s to the left. Find their common velocity after the collision.

6

m/s to the rightB.

2

m/s to the rightC.

0.5

m/s to the rightD.

1

m/s to the left

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Question 11

A particle of mass $m$ falls under gravity through a medium with quadratic drag $F_{d r a g} = - c v^{2}$ . What is the terminal velocity $v_{T}$ ?

\frac{m g}{c}

\frac{m g}{c}

\frac{c}{m g}

\frac{c}{m g}

Question 12

A particle has position $r (t) = t^{2} \hat{i} + 2 t^{3} \hat{j}$ m. What is $∣ a ∣$ at $t = 1$ s?

12

m/s²B.

14

m/s²C.

237

m/s²D.

40

m/s²

Question 13

A particle's position is $x (t) = A sin (ω t)$ . Using calculus, find $a (t)$ and identify the relationship between $a$ and $x$ .

a (t) = A ω cos (ω t)

;

a = ω x

a (t) = - A ω sin (ω t)

;

a = - x

a (t) = - A ω^{2} sin (ω t)

;

a = - ω^{2} x

a (t) = A ω^{2} cos (ω t)

;

a = ω^{2} x

Question 14

A damped oscillator has equation $m \overset{x}{¨} + b \overset{x}{˙} + k x = 0$ with $b^{2} < 4 mk$ . Which statement correctly describes the motion?

A.The system oscillates at the same frequency as the undamped case.B.The amplitude decays exponentially while the system oscillates at

ω_{d} < ω_{0}

.C.The system returns to equilibrium without oscillating.D.The energy grows exponentially due to damping.

Question 15

A variable force $F (x) = 3 x^{2}$ N acts on a particle along the $x$ -axis. What is the work done from $x = 0$ to $x = 2$ m?

6

JB.

8

JC.

12

JD.

4

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Question 16

A satellite in circular orbit of radius $r$ around Earth (mass $M_{E}$ ) has total mechanical energy $E$ . Which expression is correct?

E = - \frac{G M _{E} m}{r}

E = \frac{G M _{E} m}{2 r}

E = - \frac{G M _{E} m}{2 r}

E = 0

Question 17

A 3 kg object moves from rest and reaches a speed of $4$ m/s. What is the net work done on it?

12

JB.

24

JC.

48

JD.

6

Question 18

The gravitational potential inside a uniform solid sphere of mass $M$ and radius $R$ at distance $r < R$ from the center is $Φ (r) = - \frac{GM}{2 R ^{3}} (3 R^{2} - r^{2})$ . What is the gravitational field $g (r)$ at $r < R$ ?

g = - \frac{GM}{R ^{3}} r

(toward center)B.

g = - \frac{GM}{R ^{2}}

(constant)C.

g = - \frac{GM}{r ^{2}}

(toward center)D.

g = 0

Question 19

A spring ( $k = 500$ N/m) launches a 0.2 kg ball from rest after being compressed 0.2 m. Using energy conservation, find the ball's launch speed.

5

m/sB.

50 \approx 7.07

m/sC.

100 = 10

m/sD.

200 \approx 14.1

m/s

Question 20

A torsional pendulum has restoring torque $τ = - κ θ$ where $κ$ is the torsion constant and $I$ is the moment of inertia. What is the period of oscillation?

T = 2 π \frac{κ}{I}

T = 2 π \frac{I}{κ}

T = \frac{1}{2 π} \frac{I}{κ}

T = 2 π I κ

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Question 21

A particle of mass $m = 0.5$ kg moves in a circle of radius $r = 2$ m at speed $v = 4$ m/s. What is its angular momentum about the center?

2

kg·m²/sB.

4

kg·m²/sC.

8

kg·m²/sD.

0.5

kg·m²/s

Question 22

A constant force $F = 10$ N is applied at $0°$ to a block that moves 5 m along a frictionless surface. How much work is done by $F$ ?

0

JB.

2

JC.

50

JD.

500

Question 23

A uniform solid disk of mass $M$ and radius $R$ has moment of inertia about its central axis $I = \frac{1}{2} M R^{2}$ . A net torque $τ = 4$ N·m is applied to a disk with $M = 2$ kg and $R = 1$ m. What is the angular acceleration?

2

rad/s²B.

4

rad/s²C.

8

rad/s²D.

1

rad/s²

Question 24

A particle of mass $m$ on a frictionless horizontal surface is attached to a spring (constant $k$ ) and experiences a drag force $F_{d} = - b v$ . Which differential equation governs its motion?

m \overset{x}{¨} + k x = 0

m \overset{x}{¨} - b \overset{x}{˙} + k x = 0

m \overset{x}{¨} + b \overset{x}{˙} + k x = 0

m \overset{x}{¨} + b \overset{x}{˙} - k x = 0

Question 25

A solid cylinder ( $I = \frac{1}{2} M R^{2}$ ) rolls without slipping down a frictionless incline of height $h$ . Using energy conservation, find the speed at the bottom.

2 g h

\frac{4 g h}{3}

g h

\frac{2 g h}{3}

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Question 26

The velocity of a particle is $v (t) = t^{2} - 4 t + 3$ m/s. What is the total distance traveled from $t = 0$ to $t = 4$ s?

(Hint: factor $v$ and identify where it changes sign.)

\frac{4}{3}

mB.

4

mC.

\frac{16}{3}

mD.

6

Question 27

Two blocks, $m_{1} = 3$ kg and $m_{2} = 5$ kg, are connected by a massless rope and pulled across a frictionless surface by a force $F = 16$ N applied to $m_{2}$ . What is the tension in the rope between them?

6

NB.

10

NC.

2

m/s² (wrong unit)D.

16

Question 28

A 1 kg ball moving at $6$ m/s to the right undergoes a perfectly inelastic collision with a stationary 2 kg ball. What is their combined velocity after the collision?

6

m/sB.

3

m/sC.

2

m/sD.

1

m/s

Question 29

A spring–mass system ( $k = 50$ N/m, $m = 0.5$ kg) starts at rest at $x = 0.2$ m from equilibrium. Find the time at which the speed first equals half its maximum value. ( $ω = 10$ rad/s)

t = \frac{π}{30}

sB.

t = \frac{π}{20}

sC.

t = \frac{π}{60}

sD.

t = \frac{π}{15}

Question 30

A block of mass $m$ is pushed against a vertical wall by a horizontal force $F$ (with $F > μ_{s} m g$ ). What is the minimum $F$ needed to prevent sliding?

F_{min} = m g

F_{min} = μ_{s} m g

F_{min} = \frac{m g}{μ _{s}}

F_{min} = \frac{μ _{s}}{m g}

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Question 31

Using the gravitational field $g = - \frac{GM}{r ^{2}} \overset{r}{^}$ , find the work done by gravity on mass $m$ moving from $r_{1} = 2 R_{E}$ to $r_{2} = 4 R_{E}$ .

W = \frac{G M _{E} m}{4 R _{E}}

W = - \frac{G M _{E} m}{4 R _{E}}

W = \frac{G M _{E} m}{2 R _{E}}

W = 0

Question 32

A block of mass $m$ sits on a plane inclined at angle $θ$ (frictionless). Which expression gives the acceleration along the incline?

g cos θ

g sin θ

g tan θ

g

Question 33

A satellite of mass $m$ orbits Earth (mass $M_{E}$ , radius $R_{E}$ ) at altitude $h$ above the surface in a circular orbit. What is the orbital speed?

\frac{G M _{E}}{R _{E}}

\frac{G M _{E}}{R _{E} + h}

\frac{G M _{E}}{R _{E} + h}

\frac{G M _{E} m}{R _{E} + h}

Question 34

A particle moves in the $x y$ -plane from $(0, 0)$ to $(2, 4)$ m under the force $F = (3 x^{2}) \hat{i} + (2 y) \hat{j}$ N along the straight-line path $y = 2 x$ . What is the total work done?

8

JB.

16

JC.

24

JD.

32

Question 35

A spring–mass system ( $k$ , $m$ ) undergoes SHM. At $t = 0$ , $x_{0} = 0.06$ m and $v_{0} = 0.8$ m/s. Given $ω = 10$ rad/s, find the amplitude.

0.06

mB.

0.08

mC.

0.1

mD.

0.14

AP Physics C: Mechanics Full-length practice exam 1 — Free with Answer Explanations | Test Practice Hub