Q: A block of mass $$M$$ slides up a frictional incline at angle $$\theta$$ after being struck by a bullet. If the coefficient of kinetic friction is $$\mu_k$$, what is the magnitude of the friction force acting on the block?

$$F_f = \mu_k M g \cos \theta$$

Q: After the bullet embeds in the block, the block moves up the incline and comes to rest. Which equation relates the distance $$d$$ the block travels up the slope to its initial speed $$v_0$$?

$$0 = \frac{1}{2}(m+M)v_0$^2 - (m+M)g d \sin \theta - \mu_k (m+M)g d \cos \theta$$

Q: If the block moves a distance $$d up $$the slope, how much work does the friction force do?

$$W_f = -\mu_k (m+M)g d \cos \theta$$

Q: Which general equation can be used to determine how far up the slope the block goes after the collision?

$$\text{Initial Kinetic Energy} = \text{Work by friction} + \text{Change in Potential Energy}$$

Q: A block of mass $$M is $$pushed up a frictionless incline at angle $$\theta$$ with initial speed $$v_0. $$What is the maximum height $$h$$ reached by the block?

$$h = \frac{v_0$^2}{2g \sin \theta}$$

Q: A block with an embedded bullet moves up a frictional incline. If the block moves a distance $$d up $$the slope, what is the change in gravitational potential energy?

$$\Delta U = (m+M)g d \sin \theta$$

Question 1

A bullet of mass $$m is $$fired into and becomes embedded in a block of wood of mass $$M$$ resting on a frictionless surface. Immediately after the collision, what is the speed $$v of $$the combined system?

Accepted Answer

$$v = \frac{m v_0$}{m + M}$$

Question 2

In the bullet-block collision described above, which of the following is conserved during the collision?

Accepted Answer

The momentum of the system

Question 3

A block of mass $$M$$ slides up a frictional incline at angle $$	heta$$ after being struck by a bullet. If the coefficient of kinetic friction is $$\mu_k$$, what is the magnitude of the friction force acting on the block?

Accepted Answer

$$F_f = \mu_k M g \cos 	heta$$

Question 4

After the bullet embeds in the block, the block moves up the incline and comes to rest. Which equation relates the distance $$d$$ the block travels up the slope to its initial speed $$v_0$$?

Accepted Answer

$$0 = \frac{1}{2}(m+M)v_0$^2 - (m+M)g d \sin 	heta - \mu_k (m+M)g d \cos 	heta$$

Question 5

A small ball rolls around the inside of a frictionless cone at a constant height $$y$$ and speed $$v. $$The radius of the circle is $$r$$, and the angle between the surface and the axis is $$	heta. $$What is the correct expression for the ball's speed $$v$$?

Accepted Answer

$$v = \sqrt{g r 	an 	heta}$$

Question 6

In the scenario where a block with an embedded bullet moves up a frictional incline, which energy transformation occurs as the block moves up?

Accepted Answer

Kinetic energy is transformed into potential energy and thermal energy due to friction.

Question 7

If the block moves a distance $$d up $$the slope, how much work does the friction force do?

Accepted Answer

$$W_f = -\mu_k (m+M)g d \cos 	heta$$

Question 8

Which general equation can be used to determine how far up the slope the block goes after the collision?

Accepted Answer

$$	ext{Initial Kinetic Energy} = 	ext{Work by friction} + 	ext{Change in Potential Energy}$$

Question 9

A block of mass $$M is $$pushed up a frictionless incline at angle $$	heta$$ with initial speed $$v_0. $$What is the maximum height $$h$$ reached by the block?

Accepted Answer

$$h = \frac{v_0$^2}{2g \sin 	heta}$$

Question 10

In a perfectly inelastic collision, such as a bullet embedding in a block, which of the following statements is true?

Accepted Answer

Momentum is conserved, but mechanical energy is not.

Question 11

A block with an embedded bullet slides up a rough incline and comes to rest after traveling distance $$d. $$Which of the following best describes the energy changes in the system?

Accepted Answer

Initial kinetic energy is converted into gravitational potential energy and work done against friction.

Question 12

A block of mass $$M is at $$rest at the bottom of a frictionless incline. A bullet of mass $$m$$ traveling at speed $$v_0$$ embeds in the block. What is the total momentum of the system immediately after the collision?

Accepted Answer

$$(m+M)v = m v_0$

Question 13

A block with an embedded bullet moves up a frictional incline. If the block moves a distance $$d up $$the slope, what is the change in gravitational potential energy?

Accepted Answer

$$\Delta U = (m+M)g d \sin 	heta$$

Question 14

A small ball moves in a horizontal circle inside a frictionless cone at height $$y. $$Which force provides the necessary centripetal force for the ball's circular motion?

Accepted Answer

The horizontal component of the normal force from the cone's surface

Question 15

In the block-bullet scenario, if the block moves up the incline and comes to rest, what is the final kinetic energy of the system?

Accepted Answer

Zero

Question 16

Which of the following is a correct free-body diagram for the block with embedded bullet on the incline?

Accepted Answer

Forces: gravity downward, normal force perpendicular to surface, friction parallel and opposite to motion

Question 17

If the block with embedded bullet moves up the incline and the surface is slightly curved near $$x = 0$$, which of the following statements is true?

Accepted Answer

The block's motion may be affected by the change in normal force direction at the curve.

Question 18

In the cone-ball scenario, if the ball moves at constant speed in a horizontal circle, what is the relationship between the gravitational force and the normal force?

Accepted Answer

$$N \cos 	heta = m g$$

Question 19

A block with an embedded bullet moves up a frictional incline and comes to rest. Which equation can be used to solve for the maximum height $$h$$ reached by the block?

Accepted Answer

$$\frac{1}{2}(m+M)v_0$^2 = (m+M)g h + \mu_k (m+M)g d \cos 	heta$$

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