Q: A 0.01 kg bullet is fired at 400 m/s into a 2 kg wooden block hanging from a string. The bullet embeds itself in the block. What is the speed of the block and bullet immediately after the collision? (Neglect air resistance.)

$$v = \frac{m_{\text{bullet}} v_{\text{bullet}}}{m_{\text{bullet}} + m_{\text{block}}} = \frac{0.01 \times 400}{0.01 + 2} = 1.99\ \mathrm{m/s}$$

Question 1

A 0.01 kg bullet is fired at 400 m/s into a 2 kg wooden block hanging from a string. The bullet embeds itself in the block. What is the speed of the block and bullet immediately after the collision? (Neglect air resistance.)

Accepted Answer

$$v = \frac{m_{	ext{bullet}} v_{	ext{bullet}}}{m_{	ext{bullet}} + m_{	ext{block}}} = \frac{0.01 	imes 400}{0.01 + 2} = 1.99\ \mathrm{m/s}$$

Question 2

A 3 kg ball moving at 5 m/s collides elastically with a stationary 2 kg ball. After the collision, the 3 kg ball moves at 2 m/s at 30° above the x-axis. What is the y-component of the velocity of the 2 kg ball after the collision?

Accepted Answer

$$v_{2f,y} = -1.5\ \mathrm{m/s}$$

Question 3

Which of the following best describes the center of mass of a system?

Accepted Answer

It is the point where the entire mass of the system can be considered to be concentrated for translational motion.

Question 4

A shell explodes in midair, breaking into two fragments. Which statement is true about the motion of the center of mass after the explosion?

Accepted Answer

The center of mass continues to follow the original parabolic trajectory of the shell.

Question 5

For a system of particles, the velocity of the center of mass is given by:

Accepted Answer

$$\vec{v}_{cm} = \frac{1}{M} \sum_{i=1}^n m_i \vec{v}_i$$

Question 6

A uniform solid sphere, a uniform solid cylinder, and a uniform cube all have the same mass and size. Where is the center of mass located for each object?

Accepted Answer

At the geometric center of each object.

Question 7

A rotating wheel has an angular position $$	heta$$ that increases from $$0 to$$ $$\pi/2$$ radians in 2 seconds. What is the average angular velocity?

Accepted Answer

$$\omega_{avg} = \frac{\pi/2}{2} = \frac{\pi}{4}\ \mathrm{rad/s}$$

Question 8

If a rotating object has a positive angular velocity and a negative angular acceleration, what happens to its rotational speed?

Accepted Answer

It decreases.

Question 9

A disk rotates counterclockwise with a constant angular acceleration. Which of the following statements is correct?

Accepted Answer

Both angular velocity and angular acceleration are positive.

Question 10

A wrench is tossed in the air and spins as it moves. What path does the center of mass of the wrench follow?

Accepted Answer

A parabolic trajectory, just like a point mass.

Question 11

A bullet of mass $$m is $$fired with velocity $$v_A$$ into a block of mass $$M$$ hanging from a string. The bullet embeds in the block. What is the maximum height $$h$$ the block rises after the collision?

Accepted Answer

$$h = \frac{\left(\frac{m v_A}{m+M}\$$right)^2$$}{2g}$$

Question 12

Which of the following equations correctly expresses the conservation of linear momentum in two dimensions for a collision between two objects?

Accepted Answer

$$p_{1i,x} + p_{2i,x} = p_{1f,x} + p_{2f,x}$$ and $$p_{1i,y} + p_{2i,y} = p_{1f,y} + p_{2f,y}$$

Question 13

A wheel rotates from $$\$$theta_1$$ = 0 to$$ $$\$$theta_2$$ = 2\pi$$ radians in 4 seconds. What is the average angular velocity?

Accepted Answer

$$\omega_{avg} = \frac{2\pi}{4} = 0.5\pi\ \mathrm{rad/s}$$

Question 14

Which of the following statements about angular velocity is correct?

Accepted Answer

Angular velocity is positive for counterclockwise rotation and negative for clockwise rotation.

Question 15

A system consists of two particles: one of mass 2 kg at position (1,0) m and another of mass 3 kg at position (3,0) m. Where is the center of mass?

Accepted Answer

$$x_{cm} = \frac{2 	imes 1 + 3 	imes 3}{2+3} = 2.2\ \mathrm{m}$$

Question 16

A rotating object has an angular velocity $$\omega = 4\ \mathrm{rad/s}$$ and angular acceleration $$\alpha = -2\ \mathrm{$$rad/s^2$$}. $$How long will it take for the object to come to rest?

Accepted Answer

$$t = \frac{\omega}{|\alpha|} = \frac{4}{2} = 2\ \mathrm{s}$$

Conservation of Linear Momentum and Introduction to Rotational Motion

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