Q: A football player with a mass of 80 kg is running at 5.0 m/s and is stopped in 1.0 second by a tackler. What is the original kinetic energy (KE) of the player?

$$ KE = \frac{1}{2} m v^2$ = \frac{1}{2} \times 80 \times (5.0)^2$ = 1000 \ \mathrm{J} $$

Q: A 3000 kg subway train is brought to a stop from a speed of 0.5 m/s in 0.4 m by a large spring bumper at the end of the track. What is the spring constant $$k$$?

$$ k = \frac{mv^2$}{x^2$} = \frac{3000 \times (0.5)^2$}{(0.4)^2$} = 4687.5 \ \mathrm{N/m} $$

Q: A 3000 kg subway train is brought to a stop from a speed of 10 m/s by a large spring bumper at the end of the track. If the spring constant is $$k = 4.5 \times 10^5$ \ \mathrm{N/m}$$, how much energy is lost to dissipative forces if the spring compresses 1.0 m?

$$ E_{\text{lost}} = \frac{1}{2}mv^2$ - \frac{1}{2}kx^2$ = \frac{1}{2} \times 3000 \times 10^2$ - \frac{1}{2} \times 4.5 \times 10^5$ \times 1^2$ = 150,000 - 225,000 = -75,000 \ \mathrm{J} $$

Q: A 2 kg object is dropped from a height of 10 m. Ignoring air resistance, what is its speed just before it hits the ground?

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

Q: A spring with spring constant $$k = 1000 \ \mathrm{N/m} is $$compressed by 0.2 m. What is the potential energy stored in the spring?

$$ U = \frac{1}{2}kx^2$ = \frac{1}{2} \times 1000 \times (0.2)^2$ = 20 \ \mathrm{J} $$

Q: A 5 kg object slides down a frictionless incline from a height of 4 m. What is its speed at the bottom?

$$ v = \sqrt{2gh} = \sqrt{2 \times 9.8 \times 4} = 8.86 \ \mathrm{m/s} $$

Q: A 1.5 kg ball is thrown vertically upward with a speed of 10 m/s. What is the maximum height it reaches? (Ignore air resistance.)

$$ h = \frac{v^2$}{2g} = \frac{(10)^2$}{2 \times 9.8} = 5.1 \ \mathrm{m} $$

Question 1

A football player with a mass of 80 kg is running at 5.0 m/s and is stopped in 1.0 second by a tackler. What is the original kinetic energy (KE) of the player?

Accepted Answer

$$ KE = \frac{1}{2} m v^2$ = \frac{1}{2} 	imes 80 	imes (5.0)^2$ = 1000 \ \mathrm{J} $$

Question 2

What is the average power required to stop the football player in 1.0 second?

Accepted Answer

$$ P = \frac{\Delta E}{\Delta t} = \frac{1000}{1.0} = 1000 \ \mathrm{W} $$

Question 3

A ball of mass $$m is $$dropped from a height $$y = h. $$Using conservation of mechanical energy, what is the velocity of the ball at $$y = \frac{2}{3}h$$?

Accepted Answer

$$ v = \sqrt{2g(h - \frac{2}{3}h)} = \sqrt{2g(\frac{1}{3}h)} $$

Question 4

If the same ball is at $$y = 0$$, what is its velocity just before hitting the ground?

Accepted Answer

$$ v = \sqrt{2gh} $$

Question 5

A 3000 kg subway train is brought to a stop from a speed of 0.5 m/s in 0.4 m by a large spring bumper at the end of the track. What is the spring constant $$k$$?

Accepted Answer

$$ k = \frac{mv^2$}{x^2$} = \frac{3000 	imes (0.5)^2$}{(0.4)^2$} = 4687.5 \ \mathrm{N/m} $$

Question 6

A 3000 kg subway train is brought to a stop from a speed of 10 m/s by a large spring bumper at the end of the track. If the spring constant is $$k = 4.5 	imes 10^5$ \ \mathrm{N/m}$$, how much energy is lost to dissipative forces if the spring compresses 1.0 m?

Accepted Answer

$$ E_{	ext{lost}} = \frac{1}{2}mv^2$ - \frac{1}{2}kx^2$ = \frac{1}{2} 	imes 3000 	imes 10^2$ - \frac{1}{2} 	imes 4.5 	imes 10^5$ 	imes 1^2$ = 150,000 - 225,000 = -75,000 \ \mathrm{J} $$

Question 7

Which of the following best describes the principle of conservation of mechanical energy?

Accepted Answer

The total mechanical energy (kinetic plus potential) of a system remains constant if only conservative forces act on the system.

Question 8

A 2 kg object is dropped from a height of 10 m. Ignoring air resistance, what is its speed just before it hits the ground?

Accepted Answer

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

Question 9

A spring with spring constant $$k = 1000 \ \mathrm{N/m} is $$compressed by 0.2 m. What is the potential energy stored in the spring?

Accepted Answer

$$ U = \frac{1}{2}kx^2$ = \frac{1}{2} 	imes 1000 	imes (0.2)^2$ = 20 \ \mathrm{J} $$

Question 10

A 5 kg object slides down a frictionless incline from a height of 4 m. What is its speed at the bottom?

Accepted Answer

$$ v = \sqrt{2gh} = \sqrt{2 	imes 9.8 	imes 4} = 8.86 \ \mathrm{m/s} $$

Question 11

A 1.5 kg ball is thrown vertically upward with a speed of 10 m/s. What is the maximum height it reaches? (Ignore air resistance.)

Accepted Answer

$$ h = \frac{v^2$}{2g} = \frac{(10)^2$}{2 	imes 9.8} = 5.1 \ \mathrm{m} $$

Question 12

A 2 kg block slides across a rough surface and comes to rest after moving 5 m. If the coefficient of kinetic friction is 0.2, how much work is done by friction?

Accepted Answer

$$ W = -\mu_k m g d = -0.2 	imes 2 	imes 9.8 	imes 5 = -19.6 \ \mathrm{J} $$

Question 13

A 0.5 kg ball is dropped from a height of 2 m. If it rebounds to a height of 1.5 m, how much energy was lost during the bounce?

Accepted Answer

$$ \Delta E = m g (h_1$ - h_2$) = 0.5 	imes 9.8 	imes (2 - 1.5) = 2.45 \ \mathrm{J} $$

Question 14

A 1000 kg car is moving at 20 m/s. The driver slams on the brakes and the car skids to a stop in 50 m. What is the average force of friction?

Accepted Answer

$$ F = \frac{mv^2$}{2d} = \frac{1000 	imes (20)^2$}{2 	imes 50} = 4000 \ \mathrm{N} $$

Question 15

A 0.2 kg apple falls from a tree branch 3 m above the ground. How much kinetic energy does it have just before hitting the ground?

Accepted Answer

$$ KE = mgh = 0.2 	imes 9.8 	imes 3 = 5.88 \ \mathrm{J} $$

Question 16

A 60 W light bulb is left on for 2 hours. How much energy does it consume?

Accepted Answer

$$ E = P t = 60 	imes 2 	imes 3600 = 432,000 \ \mathrm{J} $$

Work, Power, and Energy Conservation in Physics

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