Momentum and Impulse — Hard Practice Quiz

A Physics cheat sheet for Momentum and Impulse — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Formulas & key concepts

Impulse: The impulse \(\vec{J}\) is the product of the average force \(\vec{F}\) and the time interval \(\Delta t\).

$$\vec{J} = \vec{F}\Delta t$$

Linear Momentum: The linear momentum \(\vec{p}\) is the product of mass \(m\) and velocity \(\vec{v}\).

$$\vec{p} = m\vec{v}$$

Impulse-Momentum Theorem: The impulse applied to an object equals its change in momentum.

$$(\sum \vec{F})\Delta t = m\vec{v}_f - m\vec{v}_0$$

Conservation of Linear Momentum: The total linear momentum of an isolated system remains constant.

$$m_1\vec{v}_{f1} + m_2\vec{v}_{f2} = m_1\vec{v}_{01} + m_2\vec{v}_{02}$$

Center of Mass (Position): The coordinate \(x_{cm}\) of the center of mass for a two-particle system.

$$x_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}$$

Velocity of Center of Mass: The velocity \(v_{cm}\) of the center of mass for a two-particle system.

$$v_{cm} = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2}$$

Practice quiz

  1. An object of mass $m$ is initially at rest. A constant force $F$ acts on it for a time $\Delta t$, giving it a final velocity $v_f$. If the same force $F$ were applied for a time $2\Delta t$ to an object of mass $2m$ that was initially moving with velocity $-v_f/2$, what would be the final velocity of the second object?

    • $v_f/2$
    • $v_f$
    • $2v_f$
    • $0$

    Answer: $v_f/2$

  2. Two particles, $m_1$ and $m_2$, are initially at rest. An internal explosion separates them. If $m_1 = 3m_2$, and $m_1$ moves with velocity $v_1$, what is the velocity of the center of mass of the system after the explosion?

    • $0$
    • $v_1$
    • $v_1/2$
    • $-v_1$

    Answer: $0$

  3. A system consists of two particles, $m_1$ and $m_2$, located at $x_1$ and $x_2$ respectively. The center of mass is at $x_{cm}$. If $m_1$ is moved to $x_1'$ such that the center of mass remains at $x_{cm}$, what must be the new position $x_2'$ of $m_2$?

    • $\frac{m_1 (x_1 - x_1') + m_2 x_2}{m_2}$
    • $\frac{m_1 (x_1' - x_1) + m_2 x_2}{m_2}$
    • $\frac{m_1 x_1 + m_2 x_2 - x_1'}{m_2}$
    • $\frac{m_1 x_1' + m_2 x_2}{m_1 + m_2}$

    Answer: $\frac{m_1 (x_1 - x_1') + m_2 x_2}{m_2}$

  4. A $1 \text{ kg}$ object moving at $10 \text{ m/s}$ collides head-on with a $2 \text{ kg}$ object moving at $5 \text{ m/s}$ in the opposite direction. After the collision, the $1 \text{ kg}$ object reverses direction and moves at $2 \text{ m/s}$. What is the impulse experienced by the $2 \text{ kg}$ object during the collision?

    • $12 \text{ N} \cdot \text{s}$
    • $8 \text{ N} \cdot \text{s}$
    • $-12 \text{ N} \cdot \text{s}$
    • $0 \text{ N} \cdot \text{s}$

    Answer: $12 \text{ N} \cdot \text{s}$

  5. A system of two particles has a total mass $M = m_1 + m_2$. If the velocity of the center of mass is $v_{cm}$, and particle $m_1$ has velocity $v_1$, what is the velocity of particle $m_2$ in terms of $M$, $m_1$, $v_{cm}$, and $v_1$?

    • $\frac{M v_{cm} - m_1 v_1}{m_2}$
    • $\frac{M v_{cm} + m_1 v_1}{m_2}$
    • $\frac{m_1 v_1 - M v_{cm}}{m_2}$
    • $\frac{M v_{cm}}{m_1 + m_2} - v_1$

    Answer: $\frac{M v_{cm} - m_1 v_1}{m_2}$

  6. Two identical carts, each of mass $m$, are moving towards each other with speeds $v$ and $2v$ respectively. They collide and stick together. What is the velocity of their center of mass after the collision?

    • $-v/2$
    • $v/2$
    • $-v$
    • $0$

    Answer: $-v/2$

  7. A rocket of total mass $M$ (including fuel) is initially at rest. It expels a small amount of fuel of mass $\Delta m$ at a constant exhaust velocity $v_e$ relative to the rocket. If the impulse on the rocket due to this expulsion is $J$, what is the change in the rocket's velocity?

    • $J/(M - \Delta m)$
    • $J/M$
    • $J/\Delta m$
    • $J/(M + \Delta m)$

    Answer: $J/(M - \Delta m)$

  8. Consider a system of two particles, $m_1$ and $m_2$. If $m_1$ is at $x_1$ and $m_2$ is at $x_2$, and $x_{cm}$ is their center of mass. If $m_1$ is moved to $x_1 + \delta x$, by what amount must $m_2$ be moved to keep the center of mass at the same position $x_{cm}$?

    • $-\frac{m_1}{m_2} \delta x$
    • $\frac{m_1}{m_2} \delta x$
    • $-\frac{m_2}{m_1} \delta x$
    • $\frac{m_2}{m_1} \delta x$

    Answer: $-\frac{m_1}{m_2} \delta x$

  9. An object of mass $m$ is initially at rest. It is subjected to a constant net force $F$ for a time $T$. What is the magnitude of the impulse delivered to the object during the first $T/2$ of the motion, in terms of its final momentum $p_f$ after time $T$?

    • $p_f/2$
    • $p_f$
    • $2p_f$
    • $p_f/4$

    Answer: $p_f/2$

  10. Two particles of masses $m$ and $3m$ are moving towards each other with speeds $v$ and $v/3$ respectively. They undergo a perfectly inelastic collision. What is the velocity of the combined mass after the collision?

    • $0$
    • $v/2$
    • $v/4$
    • $-v/4$

    Answer: $0$

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