Q: A 2 kg block slides on a horizontal surface with a coefficient of kinetic friction $$\mu_k = 0.3. If $$the block moves 5 m before coming to rest, what was its initial speed? (Assume $$g = 9.8\ \mathrm{m/s^2$}.)$$

$$v_i = \sqrt{2 \mu_k g \Delta r} = \sqrt{2 \times 0.3 \times 9.8 \times 5} \approx 5.4\ \mathrm{m/s}$$

Q: A 0.5 kg ball is attached to a string and swung in a horizontal circle of radius 2 m at a constant speed of 4 m/s. What is the tension in the string?

$$T = m a_c = m \frac{v^2$}{r} = 0.5 \times \frac{16}{2} = 4\ \mathrm{N}$$

Q: A force $$\vec{F} = (3\hat{i} + 4\hat{j})\ \mathrm{N}$$ acts on a particle that moves from $$(0,0) to$$ $$(2,3) m. $$What is the work done by the force?

$$W = \vec{F} \cdot \Delta \vec{r} = 3 \times 2 + 4 \times 3 = 6 + 12 = 18\ \mathrm{J}$$

Q: A 0.2 kg hockey puck slides across ice with an initial speed of 8 m/s. If the coefficient of kinetic friction is $$\mu_k = 0.05$$, how far does the puck travel before stopping?

$$\Delta r = \frac{v_i^2$}{2 \mu_k g} = \frac{64}{2 \times 0.05 \times 9.8} \approx 65.3\ \mathrm{m}$$

Q: A 0.1 kg ball is thrown vertically upward with an initial speed of 10 m/s. What is the maximum height reached by the ball? (Neglect air resistance.)

$$h = \frac{v_i^2$}{2g} = \frac{100}{2 \times 9.8} \approx 5.1\ \mathrm{m}$$

Q: A 2 kg block is attached to a spring ($$k = 100\ \mathrm{N/m}) $$and pulled 0.3 m from equilibrium, then released from rest. What is the block's speed as it passes through equilibrium?

$$v = \sqrt{\frac{k (\Delta x)^2$}{m}} = \sqrt{\frac{100 \times 0.09}{2}} = \sqrt{4.5} \approx 2.12\ \mathrm{m/s}$$

Question 1

A 2 kg block slides on a horizontal surface with a coefficient of kinetic friction $$\mu_k = 0.3. If $$the block moves 5 m before coming to rest, what was its initial speed? (Assume $$g = 9.8\ \mathrm{m/s^2$}.)$$

Accepted Answer

$$v_i = \sqrt{2 \mu_k g \Delta r} = \sqrt{2 	imes 0.3 	imes 9.8 	imes 5} \approx 5.4\ \mathrm{m/s}$$

Question 2

A 0.5 kg ball is attached to a string and swung in a horizontal circle of radius 2 m at a constant speed of 4 m/s. What is the tension in the string?

Accepted Answer

$$T = m a_c = m \frac{v^2$}{r} = 0.5 	imes \frac{16}{2} = 4\ \mathrm{N}$$

Question 3

A 3 kg object is dropped from a height of 10 m onto a spring with spring constant $$k = 400\ \mathrm{N/m}. $$What is the maximum compression $$\Delta s of $$the spring? (Neglect air resistance.)

Accepted Answer

$$\frac{1}{2} k (\Delta s)^2$ = m g h \implies \Delta s = \sqrt{\frac{2 m g h}{k}} = \sqrt{\frac{2 	imes 3 	imes 9.8 	imes 10}{400}} \approx 1.21\ \mathrm{m}$$

Question 4

A 1.5 kg cart moving at 2 m/s collides elastically with a stationary 2.5 kg cart. What is the velocity of the 1.5 kg cart after the collision?

Accepted Answer

$$v_{1f} = \frac{(m_1$ - m_2$)}{m_1$ + m_2$} v_{1i} = \frac{1.5 - 2.5}{1.5 + 2.5} 	imes 2 = -0.5\ \mathrm{m/s}$$

Question 5

A force $$\vec{F} = (3\hat{i} + 4\hat{j})\ \mathrm{N}$$ acts on a particle that moves from $$(0,0) to$$ $$(2,3) m. $$What is the work done by the force?

Accepted Answer

$$W = \vec{F} \cdot \Delta \vec{r} = 3 	imes 2 + 4 	imes 3 = 6 + 12 = 18\ \mathrm{J}$$

Question 6

A 0.2 kg hockey puck slides across ice with an initial speed of 8 m/s. If the coefficient of kinetic friction is $$\mu_k = 0.05$$, how far does the puck travel before stopping?

Accepted Answer

$$\Delta r = \frac{v_i^2$}{2 \mu_k g} = \frac{64}{2 	imes 0.05 	imes 9.8} \approx 65.3\ \mathrm{m}$$

Question 7

A 5 kg object moves in a circle of radius 3 m at a constant speed. If the net force toward the center is 20 N, what is the object's speed?

Accepted Answer

$$F_c = m \frac{v^2$}{r} \implies v = \sqrt{\frac{F_c r}{m}} = \sqrt{\frac{20 	imes 3}{5}} = \sqrt{12} \approx 3.46\ \mathrm{m/s}$$

Question 8

A 0.1 kg ball is thrown vertically upward with an initial speed of 10 m/s. What is the maximum height reached by the ball? (Neglect air resistance.)

Accepted Answer

$$h = \frac{v_i^2$}{2g} = \frac{100}{2 	imes 9.8} \approx 5.1\ \mathrm{m}$$

Question 9

A 2 kg block is attached to a spring ($$k = 100\ \mathrm{N/m}) $$and pulled 0.3 m from equilibrium, then released from rest. What is the block's speed as it passes through equilibrium?

Accepted Answer

$$v = \sqrt{\frac{k (\Delta x)^2$}{m}} = \sqrt{\frac{100 	imes 0.09}{2}} = \sqrt{4.5} \approx 2.12\ \mathrm{m/s}$$

Question 10

A 0.5 kg ball moving at 6 m/s collides inelastically with a 0.5 kg ball at rest. What is their final velocity after sticking together?

Accepted Answer

$$v_f = \frac{m_1$ v_{1i} + m_2$ v_{2i}}{m_1$ + m_2$} = \frac{0.5 	imes 6 + 0.5 	imes 0}{1.0} = 3\ \mathrm{m/s}$$

Question 11

A 1 kg object is acted on by a force $$F(x) = 4x N $$from $$x = 0 to$$ $$x = 2 m. $$What is the work done by this force?

Accepted Answer

$$W = \$$int_0$$^2 4x\ dx = 4 \$$int_0$$^2 x\ dx = 4 \left[ \frac{1}{2} x^2$ ight]_0^2$ = 4 	imes 2 = 8\ \mathrm{J}$$

Question 12

A 2 kg mass is moving at 3 m/s. What is its kinetic energy?

Accepted Answer

$$K = \frac{1}{2} m v^2$ = 0.5 	imes 2 	imes 9 = 9\ \mathrm{J}$$

Question 13

A 1 kg object is pushed by a constant force of 10 N over a distance of 5 m at an angle of $$60^\circ to $$the direction of motion. What is the work done by the force?

Accepted Answer

$$W = F \Delta r \cos 	heta = 10 	imes 5 	imes 0.5 = 25\ \mathrm{J}$$

Question 14

A 0.2 kg ball is dropped from a height of 2 m onto a hard floor and rebounds to a height of 1.5 m. What is the magnitude of the impulse delivered by the floor?

Accepted Answer

$$\Delta p = m (v_{up} + v_{down}) = 0.2 	imes (\sqrt{2 	imes 9.8 	imes 1.5} + \sqrt{2 	imes 9.8 	imes 2}) \approx 0.2 	imes (5.42 + 6.26) = 2.34\ \mathrm{kg\cdot m/s}$$

Question 15

A 1 kg block slides down a frictionless incline from a height of 4 m and compresses a spring ($$k = 200\ \mathrm{N/m}) at $$the bottom. What is the maximum compression of the spring?

Accepted Answer

$$\frac{1}{2} k (\Delta x)^2$ = m g h \implies \Delta x = \sqrt{\frac{2 m g h}{k}} = \sqrt{\frac{2 	imes 1 	imes 9.8 	imes 4}{200}} = \sqrt{0.392} \approx 0.626\ \mathrm{m}$$

Question 16

A 0.5 kg object is moving in a circle of radius 0.8 m at 3 m/s. What is the magnitude of its acceleration?

Accepted Answer

$$a_c = \frac{v^2$}{r} = \frac{9}{0.8} = 11.25\ \mathrm{m/s^2$}$$

Question 17

A 2 kg block is pulled at constant speed up a rough incline ($$\mu_k = 0.2) $$that rises 3 m vertically over a distance of 10 m. What is the total work done by friction?

Accepted Answer

$$W_{f_k} = -\mu_k N \Delta r = -\mu_k m g \cos 	heta 	imes 10$$; $$\cos 	heta = \frac{	ext{adjacent}}{	ext{hypotenuse}} = \frac{\sqrt{10^2$ - 3^2$}}{10} = 0.953$$; $$W_{f_k} = -0.2 	imes 2 	imes 9.8 	imes 0.953 	imes 10 \approx -37.4\ \mathrm{J}$$

Question 18

A 1 kg ball is thrown horizontally at 5 m/s from a height of 20 m. What is its speed just before hitting the ground? (Neglect air resistance.)

Accepted Answer

$$v = \sqrt{v_x^2$ + v_y^2$}$$; $$v_y = \sqrt{2 g h} = \sqrt{2 	imes 9.8 	imes 20} = 19.8\ \mathrm{m/s}$$; $$v = \sqrt{5^2$ + 19.8^2$} \approx 20.4\ \mathrm{m/s}$$

Question 19

A 0.3 kg object is acted on by a force that increases its velocity from 2 m/s to 6 m/s in 0.5 s. What is the average force applied?

Accepted Answer

$$F_{ave} = \frac{\Delta p}{\Delta t} = \frac{m (v_f - v_i)}{\Delta t} = \frac{0.3 	imes (6 - 2)}{0.5} = 2.4\ \mathrm{N}$$

Physics Study Guide: Dynamics, Work, Energy, and Momentum (Chapters 8–11)

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