Q: A ball is thrown vertically upward with an initial velocity of $$v_0 = 15$$\ \mathrm{m/s}$$. Neglecting air resistance, what is the maximum height the ball reaches above its starting point? (Use g = 9.80$$\ \mathrm{$$m/s^2$$}$$.)

$$h = \frac{v_0$^2}{2g} = \frac{(15)^2$}{2 \times 9.8} = 11.5\ \mathrm{m}$$

Q: A projectile is launched from the ground at an angle $$\theta = 30^\circ$$ above the horizontal with an initial speed $$v_0 = 20$$\ \mathrm{m/s}$$. What is the time of flight until it returns to the ground? (Neglect air resistance, g = 9.80$$\ \mathrm{$$m/s^2$$}$$.)

$$t = \frac{2 v_0$ \sin \theta}{g} = \frac{2 \times 20 \times \sin 30^\circ}{9.8} = 2.04\ \mathrm{s}$$

Q: A 5.0 kg box is pulled across a horizontal surface with a constant velocity by a force of 25 N applied at an angle of $$30^\circ$$ above the horizontal. What is the coefficient of kinetic friction between the box and the surface? ($$g = 9.80\ \mathrm{m/s^2$})$$

$$\mu_k = \frac{F \cos \theta}{mg - F \sin \theta} = \frac{25 \times 0.866}{49 - 25 \times 0.5} = 0.59$$

Q: A 2.0 kg object is dropped from rest from a height of 10 m. What is its speed just before it hits the ground? (Neglect air resistance, $$g = 9.80\ \mathrm{m/s^2$})$$

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

Q: A block of mass 3.0 kg slides down a frictionless incline of height 5.0 m. What is its speed at the bottom of the incline? ($$g = 9.80\ \mathrm{m/s^2$})$$

$$v = \sqrt{2gh} = \sqrt{2 \times 9.8 \times 5.0} = 9.9\ \mathrm{m/s}$$

Q: A 10 kg crate is pushed up a rough incline (angle $$30^\circ) at $$constant speed. The coefficient of kinetic friction is $$\mu_k = 0.20. $$What is the magnitude of the force required to push the crate up the incline at constant speed? ($$g = 9.80\ \mathrm{m/s^2$})$$

$$F = mg \sin \theta + \mu_k mg \cos \theta = 10 \times 9.8 \times 0.5 + 0.2 \times 10 \times 9.8 \times 0.866 = 49 + 16.97 = 65.97\ \mathrm{N}$$

Q: A 0.5 kg mass is attached to a spring with spring constant $$k = 200\ \mathrm{N/m}$$ and is pulled so that the spring is stretched by 0.10 m from its equilibrium position. What is the potential energy stored in the spring?

$$U = \frac{1}{2} k x^2$ = \frac{1}{2} \times 200 \times (0.10)^2$ = 1.0\ \mathrm{J}$$

Q: A 1.0 kg object is moving at $$3.0\ \mathrm{m/s}. $$What is its kinetic energy?

$$K = \frac{1}{2} m v^2$ = \frac{1}{2} \times 1.0 \times (3.0)^2$ = 4.5\ \mathrm{J}$$

Q: A 2.0 kg block slides across a horizontal surface with a coefficient of kinetic friction $$\mu_k = 0.30. If $$the block travels 5.0 m before coming to rest, what was its initial speed? ($$g = 9.80\ \mathrm{m/s^2$})$$

$$v_0$$ = \sqrt{2 \mu_k g d} = \sqrt{2 \times 0.30 \times 9.8 \times 5.0} = 5.4\ \mathrm{m/s}$$

Q: A 1.5 kg object is attached to a light string and swung in a horizontal circle of radius 2.0 m at a constant speed of 4.0 m/s. What is the tension in the string?

$$T = m \frac{v^2$}{r} = 1.5 \times \frac{(4.0)^2$}{2.0} = 12\ \mathrm{N}$$

Question 1

A ball is thrown vertically upward with an initial velocity of $$v_0 = 15$$\ \mathrm{m/s}$$. Neglecting air resistance, what is the maximum height the ball reaches above its starting point? (Use g = 9.80$$\ \mathrm{$$m/s^2$$}$$.)

Accepted Answer

$$h = \frac{v_0$^2}{2g} = \frac{(15)^2$}{2 	imes 9.8} = 11.5\ \mathrm{m}$$

Question 2

A projectile is launched from the ground at an angle $$	heta = 30^\circ$$ above the horizontal with an initial speed $$v_0 = 20$$\ \mathrm{m/s}$$. What is the time of flight until it returns to the ground? (Neglect air resistance, g = 9.80$$\ \mathrm{$$m/s^2$$}$$.)

Accepted Answer

$$t = \frac{2 v_0$ \sin 	heta}{g} = \frac{2 	imes 20 	imes \sin 30^\circ}{9.8} = 2.04\ \mathrm{s}$$

Question 3

A 5.0 kg box is pulled across a horizontal surface with a constant velocity by a force of 25 N applied at an angle of $$30^\circ$$ above the horizontal. What is the coefficient of kinetic friction between the box and the surface? ($$g = 9.80\ \mathrm{m/s^2$})$$

Accepted Answer

$$\mu_k = \frac{F \cos 	heta}{mg - F \sin 	heta} = \frac{25 	imes 0.866}{49 - 25 	imes 0.5} = 0.59$$

Question 4

A 2.0 kg object is dropped from rest from a height of 10 m. What is its speed just before it hits the ground? (Neglect air resistance, $$g = 9.80\ \mathrm{m/s^2$})$$

Accepted Answer

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

Question 5

A block of mass 3.0 kg slides down a frictionless incline of height 5.0 m. What is its speed at the bottom of the incline? ($$g = 9.80\ \mathrm{m/s^2$})$$

Accepted Answer

$$v = \sqrt{2gh} = \sqrt{2 	imes 9.8 	imes 5.0} = 9.9\ \mathrm{m/s}$$

Question 6

A 10 kg crate is pushed up a rough incline (angle $$30^\circ) at $$constant speed. The coefficient of kinetic friction is $$\mu_k = 0.20. $$What is the magnitude of the force required to push the crate up the incline at constant speed? ($$g = 9.80\ \mathrm{m/s^2$})$$

Accepted Answer

$$F = mg \sin 	heta + \mu_k mg \cos 	heta = 10 	imes 9.8 	imes 0.5 + 0.2 	imes 10 	imes 9.8 	imes 0.866 = 49 + 16.97 = 65.97\ \mathrm{N}$$

Question 7

A 0.5 kg mass is attached to a spring with spring constant $$k = 200\ \mathrm{N/m}$$ and is pulled so that the spring is stretched by 0.10 m from its equilibrium position. What is the potential energy stored in the spring?

Accepted Answer

$$U = \frac{1}{2} k x^2$ = \frac{1}{2} 	imes 200 	imes (0.10)^2$ = 1.0\ \mathrm{J}$$

Question 8

A 1.0 kg object is moving at $$3.0\ \mathrm{m/s}. $$What is its kinetic energy?

Accepted Answer

$$K = \frac{1}{2} m v^2$ = \frac{1}{2} 	imes 1.0 	imes (3.0)^2$ = 4.5\ \mathrm{J}$$

Question 9

A 2.0 kg block slides across a horizontal surface with a coefficient of kinetic friction $$\mu_k = 0.30. If $$the block travels 5.0 m before coming to rest, what was its initial speed? ($$g = 9.80\ \mathrm{m/s^2$})$$

Accepted Answer

$$v_0$$ = \sqrt{2 \mu_k g d} = \sqrt{2 	imes 0.30 	imes 9.8 	imes 5.0} = 5.4\ \mathrm{m/s}$$

Question 10

A 1.5 kg object is attached to a light string and swung in a horizontal circle of radius 2.0 m at a constant speed of 4.0 m/s. What is the tension in the string?

Accepted Answer

$$T = m \frac{v^2$}{r} = 1.5 	imes \frac{(4.0)^2$}{2.0} = 12\ \mathrm{N}$$

Question 11

A 0.2 kg ball is dropped from a height of 2.0 m. How much work does gravity do on the ball as it falls? ($$g = 9.80\ \mathrm{m/s^2$})$$

Accepted Answer

$$W = mgh = 0.2 	imes 9.8 	imes 2.0 = 3.92\ \mathrm{J}$$

Question 12

A 60 kg skier starts from rest at the top of a 40 m high hill. If there is no friction, what is her speed at the bottom? ($$g = 9.80\ \mathrm{m/s^2$})$$

Accepted Answer

$$v = \sqrt{2gh} = \sqrt{2 	imes 9.8 	imes 40} = 28\ \mathrm{m/s}$$

Question 13

A 1000 kg car accelerates from rest to 20 m/s in 10 seconds. What is the average power delivered to the car? (Neglect friction and air resistance.)

Accepted Answer

$$P = \frac{\Delta K}{t} = \frac{\frac{1}{2} m v^2$}{t} = \frac{0.5 	imes 1000 	imes 400}{10} = 20,000\ \mathrm{W}$$

Question 14

A 2.0 kg block is attached to a spring with $$k = 100\ \mathrm{N/m}$$ and is compressed 0.20 m from equilibrium. If released from rest, what is its speed as it passes through equilibrium?

Accepted Answer

$$v = \sqrt{\frac{k x^2$}{m}} = \sqrt{\frac{100 	imes 0.04}{2.0}} = 1.41\ \mathrm{m/s}$$

Question 15

A 0.5 kg object moves in a circle of radius 0.8 m at a constant speed of 6.0 m/s. What is the magnitude of the net force acting on the object?

Accepted Answer

$$F = m \frac{v^2$}{r} = 0.5 	imes \frac{36}{0.8} = 22.5\ \mathrm{N}$$

Question 16

A 3.0 kg block is at rest on a horizontal surface. The coefficient of static friction is $$\mu_s = 0.40. $$What is the minimum horizontal force required to start moving the block? ($$g = 9.80\ \mathrm{m/s^2$})$$

Accepted Answer

$$F = \mu_s mg = 0.40 	imes 3.0 	imes 9.8 = 11.76\ \mathrm{N}$$

Question 17

A 2.0 kg block is attached to a rope and hangs vertically. What is the tension in the rope if the block is at rest?

Accepted Answer

$$T = mg = 2.0 	imes 9.8 = 19.6\ \mathrm{N}$$

Free-Falling Objects and Linear Free Fall: Study Notes

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