Free AP Physics C: Mechanics Practice Exam (PDF)

This free AP Physics C: Mechanics practice exam tests the calculus the course is built on: integrating a time-dependent acceleration, reading stability from the derivative of a potential-energy function, locating the center of mass of a non-uniform rod, and deriving a moment of inertia from its definition. FRQ 1 sets up and solves the differential equation for a boat slowing under velocity-dependent drag; FRQ 2 combines moment of inertia, angular momentum conservation and energy for a putty-and-rod collision, where students lose marks by conserving linear momentum through a pivot.

1. Question 11 mark · Kinematics
   A particle moves along the x-axis with acceleration $a(t) = 6t$ (SI units). At $t = 0$ it is at $x = 0$ with velocity $+2$ m/s. Where is the particle at $t = 2$ s?
   • A.
     $x = 8$ m
   • B.
     $x = 12$ m
   • C.
     $x = 14$ m
   • D.
     $x = 16$ m
2. Question 21 mark · Newton's laws and drag
   A ball is thrown straight up and later returns to the launch height. Throughout the flight it experiences a drag force of magnitude $bv$ opposing its motion. Let $t_{up}$ be the time to reach the highest point and $t_{down}$ the time to fall back. Which statement is correct?
   • A.
     $t_{up} < t_{down}$, and the ball returns slower than it was launched
   • B.
     $t_{up} > t_{down}$, and the ball returns slower than it was launched
   • C.
     $t_{up} = t_{down}$, and the ball returns slower than it was launched
   • D.
     $t_{up} < t_{down}$, and the ball returns at the launch speed
3. Question 31 mark · Work and energy
   A 2.0 kg object starts from rest at $x = 0$ and moves along the x-axis under a single force $F(x) = 3x^2$, with $F$ in newtons and $x$ in metres. What is its speed at $x = 2.0$ m?
   • A.
     2.0 m/s
   • B.
     $2\sqrt{2}$ m/s
   • C.
     4.0 m/s
   • D.
     8.0 m/s
4. Question 41 mark · Work and energy
   A particle moves along the x-axis with potential energy $U(x) = x^3 - 12x$, with $U$ in joules and $x$ in metres. At which position is the particle in STABLE equilibrium?
   • A.
     $x = -2$ m
   • B.
     $x = 0$
   • C.
     $x = 2$ m
   • D.
     $x = 4$ m
5. Question 51 mark · Center of mass
   A thin rod of length $L$ lies along the x-axis from $x = 0$ to $x = L$. Its linear mass density increases linearly from zero at $x = 0$: $\lambda(x) = \lambda_0 x/L$. Where is the center of mass of the rod?
   • A.
     $x = L/3$
   • B.
     $x = L/2$
   • C.
     $x = 2L/3$
   • D.
     $x = 3L/4$
6. Question 61 mark · Momentum and collisions
   A block of mass $m$ moving at speed $v$ on a frictionless surface makes a head-on elastic collision with a block of mass $3m$ at rest. What is the speed of the $3m$ block after the collision?
   • A.
     $v/4$
   • B.
     $v/3$
   • C.
     $v/2$
   • D.
     $2v/3$
7. Question 71 mark · Rotation
   A uniform solid sphere ($I = \frac{2}{5}MR^2$) rolls without slipping from rest down an incline, descending a vertical height $h$. What is the speed of its center of mass at the bottom?
   • A.
     $\sqrt{gh}$
   • B.
     $\sqrt{10gh/7}$
   • C.
     $\sqrt{4gh/3}$
   • D.
     $\sqrt{2gh}$
8. Question 81 mark · Rotation
   A block of mass $m$ hangs from a light string wrapped around a uniform solid cylinder of mass $M$ and radius $R$ ($I = \frac{1}{2}MR^2$) that turns on a frictionless horizontal axle. The system is released from rest. What is the downward acceleration of the block?
   • A.
     $g$
   • B.
     $\dfrac{mg}{m + M}$
   • C.
     $\dfrac{2mg}{2m + M}$
   • D.
     $\dfrac{mg}{m + 2M}$
9. Question 91 mark · Angular momentum
   A 2.0 kg particle moves with constant velocity 4.0 m/s in the $+x$ direction along the line $y = 3.0$ m. Its angular momentum about the origin:
   • A.
     has magnitude $24\ \text{kg}\cdot\text{m}^2\text{/s}$ and is constant
   • B.
     has magnitude $24\ \text{kg}\cdot\text{m}^2\text{/s}$ and increases as the particle moves away from the origin
   • C.
     is zero, because the particle moves in a straight line
   • D.
     has magnitude $8.0\ \text{kg}\cdot\text{m}^2\text{/s}$ and is constant
10. Question 101 mark · Oscillations
    A 0.50 kg object on a frictionless surface experiences a net force $F = -32x$, with $F$ in newtons and $x$ in metres. What is the period of its oscillation?
    • A.
      $\pi/8$ s
    • B.
      $\pi/4$ s
    • C.
      $\pi/2$ s
    • D.
      $\pi$ s
11. Question 111 mark · Gravitation
    A satellite of mass $m$ is in a circular orbit of radius $r$ around a planet of mass $M$. What minimum work must be done on the satellite to move it to a circular orbit of radius $2r$?
    • A.
      $\dfrac{GMm}{8r}$
    • B.
      $\dfrac{GMm}{4r}$
    • C.
      $\dfrac{GMm}{2r}$
    • D.
      $\dfrac{GMm}{r}$
12. Question 121 mark · Gravitation
    The escape speed from the surface of Earth is about 11.2 km/s. A planet has twice Earth's mass and twice Earth's radius. What is the escape speed from its surface?
    • A.
      5.6 km/s
    • B.
      7.9 km/s
    • C.
      11.2 km/s
    • D.
      22.4 km/s
13. Question 1312 marks · Newton's laws and drag
    Drag. A motorboat of mass $m = 200$ kg is moving at $v_0 = 6.0$ m/s across still water when its engine is shut off at $t = 0$. The water exerts a resistive force on the hull of magnitude $bv$, directed opposite to the velocity, where $b = 40$ kg/s. Ignore all other horizontal forces.
    (a)

    Write Newton's second law for the boat as a differential equation for its speed $v(t)$, and give the numerical value of the constant that appears.

    [2]
    (b)

    Solve the differential equation to obtain $v(t)$, showing the separation of variables and the use of the initial condition.

    [3]
    (c)

    Calculate the time at which the boat's speed has fallen to half its initial value.

    [2]
    (d)

    Calculate the total distance the boat travels after the engine is shut off (that is, as $t \to \infty$).

    [3]
    (e)

    Suppose instead that the resistive force had magnitude $cv^2$, where $c$ is a positive constant. Using the differential equation (you need not solve for $v(t)$), argue whether the boat would travel a finite or an unbounded distance before its speed reaches zero.

    [2]
14. Question 1412 marks · Rotation
    Rod and putty. A thin uniform rod of mass $M = 1.5$ kg and length $L = 1.0$ m hangs at rest from a frictionless pivot at its upper end and is free to swing in a vertical plane. A lump of putty of mass $m = 0.50$ kg moving horizontally at $v_0 = 4.0$ m/s strikes the lower end of the rod and sticks to it. Use $g = 9.8\ \text{m/s}^2$.
    (a)

    Starting from $I = \int r^2\,dm$, derive the rotational inertia of the rod about the pivot. Then calculate the total rotational inertia of the rod-and-putty system about the pivot after the collision.

    [3]
    (b)

    Calculate the angular speed of the rod-and-putty system immediately after the collision.

    [3]
    (c)

    Explain why angular momentum about the pivot is conserved during the collision, but the linear momentum of the rod-and-putty system is not.

    [2]
    (d)

    Calculate the maximum angle, measured from the vertical, through which the rod swings after the collision.

    [4]

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