Q: A ball is thrown vertically upward with an initial velocity of $$v_0 = 12.0$$\ \mathrm{m/s}$$. Neglecting air resistance, what is the maximum height reached by the ball? (Use g = 9.80$$\ \mathrm{$$m/s^2$$}$$)

$$h = \dfrac{v_0$^2}{2g} = \dfrac{(12.0)^2$}{2 \times 9.80} = 7.35\ \mathrm{m}$$

Q: A monkey drops from a tree limb 1.80 m above the ground. How long does it take to reach the ground? (Neglect air resistance, $$g = 9.80\ \mathrm{m/s^2$})$$

$$t = \sqrt{\dfrac{2h}{g}} = \sqrt{\dfrac{2 \times 1.80}{9.80}} = 0.606\ \mathrm{s}$$

Q: A projectile is launched at an angle $$\theta$$ above the horizontal with initial speed $$v_0. $$What is the formula for its horizontal range $$R ($$assuming it lands at the same height it was launched)?

$$R = \dfrac{v_0$^2 \sin(2\theta)}{g}$$

Q: A 100 kg crate is pushed up a $$30^\circ$$ ramp at constant speed. What is the magnitude of the normal force exerted by the ramp on the crate?

$$N = mg \cos(\theta) = 100 \times 9.80 \times \cos(30^\circ) = 849\ \mathrm{N}$$

Q: A 32.0-kg penguin slides down a $$30.0^\circ$$ icy incline with a coefficient of kinetic friction $$\mu_k = 0.350. $$What is the penguin's acceleration?

$$a = g \sin(\theta) - \mu_k g \cos(\theta) = 9.80 \sin(30.0^\circ) - 0.350 \times 9.80 \cos(30.0^\circ) = 4.90 - 2.97 = 1.93\ \mathrm{m/s^2$}$$

Q: A 1.0 kg object is dropped from rest. What is its velocity after falling 5.0 m? ($$g = 9.80\ \mathrm{m/s^2$})$$

$$v = \sqrt{2gh} = \sqrt{2 \times 9.80 \times 5.0} = 9.90\ \mathrm{m/s}$$

Q: A block is pulled across a horizontal surface with a force of $$F = 85.0\ \mathrm{N} at an $$angle of $$5.00^\circ$$ above the horizontal. The block has a weight of $$225\ \mathrm{N}$$ and the coefficient of kinetic friction is $$\mu_k = 0.250. $$What is the magnitude of the friction force?

$$f_k = \mu_k N = 0.250 \times (225 - 85.0 \sin(5.00^\circ)) = 0.250 \times 218.6 = 54.7\ \mathrm{N}$$

Q: A 0.500 kg mass hangs at the midpoint of a wire stretched between two poles 50.0 m apart, causing the wire to sag by 0.150 m. What is the tension in the wire?

$$T = \dfrac{mg}{2 \sin \theta}$$, where $$\sin \theta \approx \dfrac{0.150}{25.0} = 0.006$$; $$T = \dfrac{0.500 \times 9.80}{2 \times 0.006} = 408\ \mathrm{N}$$

Question 1

A ball is thrown vertically upward with an initial velocity of $$v_0 = 12.0$$\ \mathrm{m/s}$$. Neglecting air resistance, what is the maximum height reached by the ball? (Use g = 9.80$$\ \mathrm{$$m/s^2$$}$$)

Accepted Answer

$$h = \dfrac{v_0$^2}{2g} = \dfrac{(12.0)^2$}{2 	imes 9.80} = 7.35\ \mathrm{m}$$

Question 2

A monkey drops from a tree limb 1.80 m above the ground. How long does it take to reach the ground? (Neglect air resistance, $$g = 9.80\ \mathrm{m/s^2$})$$

Accepted Answer

$$t = \sqrt{\dfrac{2h}{g}} = \sqrt{\dfrac{2 	imes 1.80}{9.80}} = 0.606\ \mathrm{s}$$

Question 3

A springbok leaps upward with an initial speed of $$6.00\ \mathrm{m/s}. $$What is the time taken to reach its maximum height? ($$g = 9.80\ \mathrm{m/s^2$})$$

Accepted Answer

$$t = \dfrac{v_0$}{g} = \dfrac{6.00}{9.80} = 0.612\ \mathrm{s}$$

Question 4

A projectile is launched at an angle $$	heta$$ above the horizontal with initial speed $$v_0. $$What is the formula for its horizontal range $$R ($$assuming it lands at the same height it was launched)?

Accepted Answer

$$R = \dfrac{v_0$^2 \sin(2	heta)}{g}$$

Question 5

A 100 kg crate is pushed up a $$30^\circ$$ ramp at constant speed. What is the magnitude of the normal force exerted by the ramp on the crate?

Accepted Answer

$$N = mg \cos(	heta) = 100 	imes 9.80 	imes \cos(30^\circ) = 849\ \mathrm{N}$$

Question 6

A block of mass $$m is $$connected to another block of mass $$M by a $$light string over a frictionless pulley (Atwood's machine, $$M \u003e m). $$What is the acceleration of the system?

Accepted Answer

$$a = \dfrac{(M - m)g}{M + m}$$

Question 7

A 32.0-kg penguin slides down a $$30.0^\circ$$ icy incline with a coefficient of kinetic friction $$\mu_k = 0.350. $$What is the penguin's acceleration?

Accepted Answer

$$a = g \sin(	heta) - \mu_k g \cos(	heta) = 9.80 \sin(30.0^\circ) - 0.350 	imes 9.80 \cos(30.0^\circ) = 4.90 - 2.97 = 1.93\ \mathrm{m/s^2$}$$

Question 8

A 1.0 kg object is dropped from rest. What is its velocity after falling 5.0 m? ($$g = 9.80\ \mathrm{m/s^2$})$$

Accepted Answer

$$v = \sqrt{2gh} = \sqrt{2 	imes 9.80 	imes 5.0} = 9.90\ \mathrm{m/s}$$

Question 9

A block is pulled across a horizontal surface with a force of $$F = 85.0\ \mathrm{N} at an $$angle of $$5.00^\circ$$ above the horizontal. The block has a weight of $$225\ \mathrm{N}$$ and the coefficient of kinetic friction is $$\mu_k = 0.250. $$What is the magnitude of the friction force?

Accepted Answer

$$f_k = \mu_k N = 0.250 	imes (225 - 85.0 \sin(5.00^\circ)) = 0.250 	imes 218.6 = 54.7\ \mathrm{N}$$

Question 10

A 0.500 kg mass hangs at the midpoint of a wire stretched between two poles 50.0 m apart, causing the wire to sag by 0.150 m. What is the tension in the wire?

Accepted Answer

$$T = \dfrac{mg}{2 \sin 	heta}$$, where $$\sin 	heta \approx \dfrac{0.150}{25.0} = 0.006$$; $$T = \dfrac{0.500 	imes 9.80}{2 	imes 0.006} = 408\ \mathrm{N}$$

Question 11

A 60.0-kg woman runs up a 3.0 m high flight of stairs in 4.0 s, starting from rest and reaching a final speed of 2.0 m/s. What is her average power output?

Accepted Answer

$$P = \dfrac{mgh + \frac{1}{2}mv^2$}{t} = \dfrac{60.0 	imes 9.80 	imes 3.0 + 0.5 	imes 60.0 	imes (2.0)^2$}{4.0} = \dfrac{1,764 + 120}{4.0} = 471\ \mathrm{W}$$

Question 12

A spring with spring constant $$k = 860\ \mathrm{N/m} is $$stretched $$x = 0.47\ \mathrm{m}$$ from equilibrium. What is the elastic potential energy stored in the spring?

Accepted Answer

$$U = \dfrac{1}{2}kx^2$ = \dfrac{1}{2} 	imes 860 	imes (0.47)^2$ = 94.9\ \mathrm{J}$$

Question 13

A car of mass $$m$$ moves at constant speed $$v on a $$straight, dry road. What is the direction of the friction force acting on the car?

Accepted Answer

Opposite to the direction of motion

Question 14

A block of mass $$m$$ slides on a frictionless surface and is brought to rest by a constant force $$F$$ over a distance $$d. $$What is the work done by the force?

Accepted Answer

$$W = -Fd$$

Question 15

A 1.0 kg object is lifted vertically upward by 2.0 m. What is the change in its gravitational potential energy? ($$g = 9.80\ \mathrm{m/s^2$})$$

Accepted Answer

$$\Delta U = mgh = 1.0 	imes 9.80 	imes 2.0 = 19.6\ \mathrm{J}$$

Question 16

A 2.0 kg block is moving at $$3.0\ \mathrm{m/s} on a $$horizontal surface. What is its kinetic energy?

Accepted Answer

$$K = \dfrac{1}{2}mv^2$ = \dfrac{1}{2} 	imes 2.0 	imes (3.0)^2$ = 9.0\ \mathrm{J}$$

Question 17

A block of mass $$m is $$pulled up a ramp of height $$h$$ and length $$d at $$constant speed. What is the minimum work required to move the block up the ramp (neglecting friction)?

Accepted Answer

$$W = mgh$$

Free-Falling Objects and Acceleration Due to Gravity

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