Q: A projectile is launched at an angle of $$30^\circ$$ above the horizontal with an initial speed of $$40\ \mathrm{m/s}. $$What is the vertical component of its initial velocity?

$$v_{0,y} = v_0$ \sin(\theta) = 40 \sin(30^\circ) = 20\ \mathrm{m/s}$$

Q: A $$2\ \mathrm{kg}$$ block is pulled across a horizontal surface with a coefficient of kinetic friction $$\mu_k = 0.3. $$What is the magnitude of the frictional force?

$$f_k = \mu_k N = 0.3 \times 2 \times 9.8 = 5.88\ \mathrm{N}$$

Q: A $$5\ \mathrm{kg}$$ object is dropped from a height of $$10\ \mathrm{m}. $$What is its velocity just before hitting the ground? (Ignore air resistance.)

$$v = \sqrt{2gh} = \sqrt{2 \times 9.8 \times 10} = \sqrt{196} = 14\ \mathrm{m/s}$$

Q: A $$0.5\ \mathrm{kg}$$ ball is attached to a string and swung in a horizontal circle of radius $$2\ \mathrm{m} at a $$speed of $$4\ \mathrm{m/s}. $$What is the centripetal force acting on the ball?

$$F_c = \frac{mv^2$}{r} = \frac{0.5 \times 4^2$}{2} = \frac{0.5 \times 16}{2} = 4\ \mathrm{N}$$

Q: A spring with a constant $$k = 200\ \mathrm{N/m} is $$compressed by $$0.1\ \mathrm{m}. $$What is the potential energy stored in the spring?

$$PE_s = \frac{1}{2} k x^2$ = \frac{1}{2} \times 200 \times (0.1)^2$ = 1\ \mathrm{J}$$

Q: A $$10\ \mathrm{kg}$$ object is placed in water ($$\rho_w = 1000\ \mathrm{kg/m^3$}) $$and displaces $$0.01\ \mathrm{m^3$} of $$water. What is the buoyant force acting on the object?

$$F_B = \rho_{fluid} V_{fluid} g = 1000 \times 0.01 \times 9.8 = 98\ \mathrm{N}$$

Question 1

A car accelerates from rest to a speed of $$20\ \mathrm{m/s} in$$ $$5\ \mathrm{s}. $$What is the average acceleration of the car?

Accepted Answer

$$a_{av} = \frac{v_f - v_i}{t_f - t_i} = \frac{20 - 0}{5} = 4\ \mathrm{m/s^2$}$$

Question 2

A projectile is launched at an angle of $$30^\circ$$ above the horizontal with an initial speed of $$40\ \mathrm{m/s}. $$What is the vertical component of its initial velocity?

Accepted Answer

$$v_{0,y} = v_0$ \sin(	heta) = 40 \sin(30^\circ) = 20\ \mathrm{m/s}$$

Question 3

A $$2\ \mathrm{kg}$$ block is pulled across a horizontal surface with a coefficient of kinetic friction $$\mu_k = 0.3. $$What is the magnitude of the frictional force?

Accepted Answer

$$f_k = \mu_k N = 0.3 	imes 2 	imes 9.8 = 5.88\ \mathrm{N}$$

Question 4

A $$5\ \mathrm{kg}$$ object is dropped from a height of $$10\ \mathrm{m}. $$What is its velocity just before hitting the ground? (Ignore air resistance.)

Accepted Answer

$$v = \sqrt{2gh} = \sqrt{2 	imes 9.8 	imes 10} = \sqrt{196} = 14\ \mathrm{m/s}$$

Question 5

A $$0.5\ \mathrm{kg}$$ ball is attached to a string and swung in a horizontal circle of radius $$2\ \mathrm{m} at a $$speed of $$4\ \mathrm{m/s}. $$What is the centripetal force acting on the ball?

Accepted Answer

$$F_c = \frac{mv^2$}{r} = \frac{0.5 	imes 4^2$}{2} = \frac{0.5 	imes 16}{2} = 4\ \mathrm{N}$$

Question 6

A spring with a constant $$k = 200\ \mathrm{N/m} is $$compressed by $$0.1\ \mathrm{m}. $$What is the potential energy stored in the spring?

Accepted Answer

$$PE_s = \frac{1}{2} k x^2$ = \frac{1}{2} 	imes 200 	imes (0.1)^2$ = 1\ \mathrm{J}$$

Question 7

A $$1\ \mathrm{kg}$$ object moving at $$3\ \mathrm{m/s}$$ collides with a stationary $$2\ \mathrm{kg}$$ object. If the collision is perfectly elastic, what is the velocity of the $$1\ \mathrm{kg}$$ object after the collision?

Accepted Answer

$$v_1$$' = \frac{$$m_1$$ - $$m_2$$}{$$m_1$$ + $$m_2$$} $$v_1$$ + \frac{2m_2}{$$m_1$$ + $$m_2$$} $$v_2$$ = \frac{1 - 2}{1 + 2} 	imes 3 + \frac{2 	imes 2}{1 + 2} 	imes 0 = -1\ \mathrm{m/s}$$

Question 8

A $$10\ \mathrm{kg}$$ object is placed in water ($$\rho_w = 1000\ \mathrm{kg/m^3$}) $$and displaces $$0.01\ \mathrm{m^3$} of $$water. What is the buoyant force acting on the object?

Accepted Answer

$$F_B = ho_{fluid} V_{fluid} g = 1000 	imes 0.01 	imes 9.8 = 98\ \mathrm{N}$$

Question 9

A $$0.2\ \mathrm{kg}$$ mass attached to a spring oscillates with a frequency of $$2\ \mathrm{Hz}. $$What is the spring constant $$k$$?

Accepted Answer

$$f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \Rightarrow k = (2\pi f)^2$ m = (2\pi 	imes 2)^2$ 	imes 0.2 = (4\pi)^2$ 	imes 0.2 = 16\pi^2$ 	imes 0.2 \approx 31.6\ \mathrm{N/m}$$

Question 10

A $$0.5\ \mathrm{m}$$ long pendulum swings with small amplitude. What is its period?

Accepted Answer

$$T = 2\pi \sqrt{\frac{L}{g}} = 2\pi \sqrt{\frac{0.5}{9.8}} \approx 1.42\ \mathrm{s}$$

Question 11

A string of length $$1\ \mathrm{m}$$ and mass $$0.01\ \mathrm{kg} is $$under tension of $$100\ \mathrm{N}. $$What is the speed of a wave on the string?

Accepted Answer

$$v_w = \sqrt{\frac{F}{\mu}}$$, $$\mu = \frac{m}{L} = \frac{0.01}{1} = 0.01\ \mathrm{kg/m}$$, $$v_w = \sqrt{\frac{100}{0.01}} = \sqrt{10,000} = 100\ \mathrm{m/s}$$

Question 12

Two sound sources emit frequencies of $$440\ \mathrm{Hz}$$ and $$445\ \mathrm{Hz}. $$What is the beat frequency heard?

Accepted Answer

$$f_B = |f_1$ - f_2$| = |440 - 445| = 5\ \mathrm{Hz}$$

Question 13

A $$0.1\ \mathrm{kg}$$ sample of water at $$20^\circ\mathrm{C} is $$heated to $$80^\circ\mathrm{C}. $$How much heat is required? ($$c = 4186\ \mathrm{J/kg\cdot K})$$

Accepted Answer

$$Q = mc\Delta T = 0.1 	imes 4186 	imes (80 - 20) = 0.1 	imes 4186 	imes 60 = 25,116\ \mathrm{J}$$

Question 14

A gas expands isobarically from $$2\ \mathrm{L} to$$ $$5\ \mathrm{L} at a $$pressure of $$1.5 	imes 10^5$\ \mathrm{Pa}. $$What is the work done by the gas?

Accepted Answer

$$W = p\Delta V = 1.5 	imes 10^5$ 	imes (5 - 2) 	imes 10$$^{-3}$$ = 1.5 	imes 10^5$ 	imes 3 	imes 10$$^{-3}$$ = 450\ \mathrm{J}$$

Question 15

A $$3\ \mathrm{kg}$$ object is moving in a circle of radius $$2\ \mathrm{m}$$ with angular velocity $$5\ \mathrm{rad/s}. $$What is its rotational kinetic energy?

Accepted Answer

$$I = mr^2$ = 3 	imes 2^2$ = 12\ \mathrm{kg\cdot m^2$}$$; $$KE_{rot} = \frac{1}{2} I \$$omega^2$$ = \frac{1}{2} 	imes 12 	imes 25 = 150\ \mathrm{J}$$

Question 16

A $$0.5\ \mathrm{kg}$$ object is dropped into a $$0.2\ \mathrm{kg}$$ container of water at $$25^\circ\mathrm{C}. $$The object is initially at $$100^\circ\mathrm{C}$$ and cools to $$25^\circ\mathrm{C}. If $$the specific heat of the object is $$900\ \mathrm{J/kg\cdot K}$$ and water is $$4186\ \mathrm{J/kg\cdot K}$$, what is the heat gained by the water?

Accepted Answer

Question 17

A $$1\ \mathrm{kg}$$ block is pushed against a spring (with $$k = 100\ \mathrm{N/m}) $$compressing it by $$0.2\ \mathrm{m}. $$The block is then released. What is its speed as it leaves the spring, assuming no friction?

Accepted Answer

Question 18

A $$2\ \mathrm{kg}$$ block slides down a $$30^\circ$$ incline that is $$5\ \mathrm{m}$$ long. What is its speed at the bottom, assuming no friction?

Accepted Answer

$$h = 5 \sin(30^\circ) = 2.5\ \mathrm{m}$$; $$v = \sqrt{2gh} = \sqrt{2 	imes 9.8 	imes 2.5} = \sqrt{49} = 7\ \mathrm{m/s}$$

Question 19

A $$0.1\ \mathrm{kg}$$ object is moving at $$10\ \mathrm{m/s}$$ and collides inelastically with a $$0.2\ \mathrm{kg}$$ object at rest. What is their final velocity?

Accepted Answer

$$v_f = \frac{m_1$ v_1$ + m_2$ v_2$}{m_1$ + m_2$} = \frac{0.1 	imes 10 + 0.2 	imes 0}{0.1 + 0.2} = \frac{1}{0.3} = 3.33\ \mathrm{m/s}$$

Question 20

A $$0.5\ \mathrm{kg}$$ block is attached to a string and lowered into a fluid with density $$1200\ \mathrm{kg/m^3$}. $$The block has a volume of $$0.0005\ \mathrm{m^3$}. $$What is the net force on the block while fully submerged?

Accepted Answer

$$F_B = ho_{fluid} V_{block} g = 1200 	imes 0.0005 	imes 9.8 = 5.88\ \mathrm{N}$$; $$F_g = m g = 0.5 	imes 9.8 = 4.9\ \mathrm{N}$$; Net force $$= F_B - F_g = 5.88 - 4.9 = 0.98\ \mathrm{N} ($$upward)

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