Q: A 1100 kg car travels at 19 m/s and slows to 12 m/s after crossing a 32 m sandy stretch. What is the magnitude of the average net force on the car in the sandy section?

$$F = \frac{m(v_f^2$ - v_i^2$)}{2d} = \frac{1100 \times (12^2$ - 19^2$)}{2 \times 32} = -\mathbf{1,320} \ \mathrm{N}$$

Q: A 6.60 kg block slides with an initial speed of 2.24 m/s up a ramp inclined at $$18.4^\circ$$ with the horizontal. The coefficient of kinetic friction is $$\mu_k = 0.44. $$Using energy conservation, what is the distance the block slides before coming to rest?

$$d = \frac{v_0$^2}{2g(\sin\theta + \mu_k \cos\theta)} = \frac{(2.24)^2$}{2 \times 9.8 (\sin 18.4^\circ + 0.44 \cos 18.4^\circ)} \approx 0.47 \ \mathrm{m}$$

Q: A child of mass 30 kg rides a skateboard along a frictionless track. If the total mechanical energy at point C is 1200 J, what is the child's speed at point A, which is 2.0 m higher than point C?

$$E_{A} = E_{C} + mgh = 1200 + 30 \times 9.8 \times 2 = 1788 \ \mathrm{J}$$, $$v_A = \sqrt{\frac{2E_A}{m}} \approx 10.9 \ \mathrm{m/s}$$

Q: A block of mass $$m is $$pulled up a rough incline at angle $$\theta$$ with a force $$F$$ parallel to the incline. The coefficient of kinetic friction is $$\mu_k. $$What is the total work done by all forces as the block moves a distance $$d up $$the incline?

$$W_{total} = Fd - m g \sin\theta \ d - \mu_k m g \cos\theta \ d$$

Q: A ball hangs on a rope of 0.8 m. The ball is pushed horizontally so that it has a speed of 1.3 m/s. What angle $$\theta$$ does the rope make with the vertical when the ball’s kinetic energy is halved?

$$\cos\theta = 1 - \frac{v^2$}{4gL}$$, \ so \ \theta \approx 15.5^\circ$

Question 1

A 0.5 kg cart moving at 2 m/s collides elastically with a stationary 0.5 kg cart on a frictionless track. After the collision, the first cart comes to rest. What is the velocity of the second cart after the collision?

Accepted Answer

2 \ \mathrm{m/s}

Question 2

A force of 10 N acts on a 2 kg object for 3 s. What is the change in the object’s momentum?

Accepted Answer

15 \ \mathrm{kg \cdot m/s}

Question 3

A 5.0 kg object is pulled horizontally across a frictionless surface by a constant force of 20 N. If the object moves 4.0 m, what is the work done by the force?

Accepted Answer

80 \ \mathrm{J}

Question 4

Which of the following statements correctly expresses the Work–Energy Theorem?

Accepted Answer

The net work done on an object equals the change in its kinetic energy.

Question 5

Two ice skaters, initially at rest, push off each other on a frictionless surface. Which of the following statements is true?

Accepted Answer

The total momentum of the system remains zero after they push apart.

Question 6

A 10 kg box is pulled on a frictionless horizontal surface by a 20 N horizontal force. What is the acceleration of the box?

Accepted Answer

2.0 \ \mathrm{$$m/s^2$$}

Question 7

A 1100 kg car travels at 19 m/s and slows to 12 m/s after crossing a 32 m sandy stretch. What is the magnitude of the average net force on the car in the sandy section?

Accepted Answer

$$F = \frac{m(v_f^2$ - v_i^2$)}{2d} = \frac{1100 	imes (12^2$ - 19^2$)}{2 	imes 32} = -\mathbf{1,320} \ \mathrm{N}$$

Question 8

A 6.60 kg block slides with an initial speed of 2.24 m/s up a ramp inclined at $$18.4^\circ$$ with the horizontal. The coefficient of kinetic friction is $$\mu_k = 0.44. $$Using energy conservation, what is the distance the block slides before coming to rest?

Accepted Answer

$$d = \frac{v_0$^2}{2g(\sin	heta + \mu_k \cos	heta)} = \frac{(2.24)^2$}{2 	imes 9.8 (\sin 18.4^\circ + 0.44 \cos 18.4^\circ)} \approx 0.47 \ \mathrm{m}$$

Question 9

A 1.2 kg block is held at rest against a spring with $$k = 730 \ \mathrm{N/m}. $$The spring is compressed by $$d$$ and released. The block slides across a frictionless surface except for a rough patch of width 5.0 cm with $$\mu_k = 0.44. If $$the block's speed after crossing the patch is 2.3 m/s, what is the required compression $$d$$?

Accepted Answer

$$\frac{1}{2} k d^2$ = \frac{1}{2} m v^2$ + \mu_k m g \Delta x \implies d = \sqrt{\frac{m v^2$ + 2 \mu_k m g \Delta x}{k}} \approx 0.22 \ \mathrm{m}$$

Question 10

A small wooden block of mass 0.800 kg is suspended from a 1.60 m cord. A 12.0 g bullet is fired horizontally and embeds in the block. After collision, the block and bullet swing up to a height of 0.800 m. If the tension in the cord at the highest point is 4.80 N, what was the initial speed of the bullet?

Accepted Answer

$$v_0$$ = \frac{(m_{block} + m_{bullet}) v_{after}}{m_{bullet}}$$, \ where \ v_{after} = \sqrt{2gh}$$, \ so \ $$v_0$$ \approx 570 \ \mathrm{m/s}$$

Question 11

Three air carts with masses $$m$$, $$2m$$, and $$4m ($$from left to right) are on a frictionless track. The first two move right with speed $$v_0$$, the third is at rest. They stick together after a completely inelastic collision. What is the final speed of the combined mass?

Accepted Answer

$$v_f = \frac{m v_0$ + 2m v_0$}{m + 2m + 4m} = \frac{3m v_0$}{7m} = \frac{3}{7} v_0$

Question 12

A child of mass 30 kg rides a skateboard along a frictionless track. If the total mechanical energy at point C is 1200 J, what is the child's speed at point A, which is 2.0 m higher than point C?

Accepted Answer

$$E_{A} = E_{C} + mgh = 1200 + 30 	imes 9.8 	imes 2 = 1788 \ \mathrm{J}$$, $$v_A = \sqrt{\frac{2E_A}{m}} \approx 10.9 \ \mathrm{m/s}$$

Question 13

A block of mass $$m is $$pulled up a rough incline at angle $$	heta$$ with a force $$F$$ parallel to the incline. The coefficient of kinetic friction is $$\mu_k. $$What is the total work done by all forces as the block moves a distance $$d up $$the incline?

Accepted Answer

$$W_{total} = Fd - m g \sin	heta \ d - \mu_k m g \cos	heta \ d$$

Question 14

A ball hangs on a rope of 0.8 m. The ball is pushed horizontally so that it has a speed of 1.3 m/s. What angle $$	heta$$ does the rope make with the vertical when the ball’s kinetic energy is halved?

Accepted Answer

$$\cos	heta = 1 - \frac{v^2$}{4gL}$$, \ so \ 	heta \approx 15.5^\circ$

Question 15

A block A of mass $$m_1 = 2.4$ kg is connected to block B of mass m_2$ = 1.8 kg $$over a frictionless pulley. Block A is on a horizontal surface with $$\mu_k = 0.45. $$When released, what is the speed of the blocks just before block B lands after falling $$d = 0.5 m$$?

Accepted Answer

$$v = \sqrt{2g d \left( \frac{m_2$ - \mu_k m_1$}{m_1$ + m_2$} \right)} \approx 1.5 \ \mathrm{m/s}$$

Question 16

Two blocks are connected by a cord over a frictionless pulley as shown (100 kg on a 30° incline, 50 kg on a 53.1° incline). Which way does the system move when released, and what is the acceleration?

Accepted Answer

The system moves toward the 100 kg block; $$a = \frac{100 	imes g \sin 30^\circ - 50 	imes g \sin 53.1^\circ}{100 + 50} \approx 0.33 \ \mathrm{m/s^2$}$$

Question 17

A 45.0 N block is pulled with a force of 45.0 N at an angle $$	heta$$ above the horizontal for 1.5 m, reaching a speed of 2.60 m/s, and 50.0 J of work is done. What is the angle $$	heta$$?

Accepted Answer

$$\cos	heta = \frac{W}{F d} = \frac{50.0}{45.0 	imes 1.5} \implies 	heta \approx 37^\circ$$

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