Q: A 30 g marble is released from rest at the top of a ramp 62 cm above a table. The table is 95 cm above the floor. The marble leaves the ramp horizontally and lands 100 cm from the edge of the table. Assuming no energy is lost to friction, what would be the theoretical speed of the marble as it leaves the ramp?

$$v = \sqrt{2gh} = \sqrt{2 \times 9.8 \ \mathrm{m/s^2$} \times 0.62 \ \mathrm{m}} = 3.49 \ \mathrm{m/s}$$

Q: A 0.030 kg marble is released from a height of 62 cm above a table and lands 100 cm from the table edge. If the table is 95 cm above the floor, what is the total mechanical energy lost to friction if the theoretical speed is 3.49 m/s but the actual speed is 3.00 m/s?

$$\Delta E = \frac{1}{2} m (v_{\text{theory}}^2 - v_{\text{actual}}^2) = \frac{1}{2} \times 0.030 \times (3.49^2$ - 3.00^2$) = 0.077 \ \mathrm{J}$$

Q: A 0.5 kg object is dropped from a height of 10 m. If it lands with a speed of 12 m/s, how much energy was lost to air resistance?

$$\Delta E = mgh - \frac{1}{2}mv^2$ = 0.5 \times 9.8 \times 10 - 0.5 \times 0.5 \times 12^2$ = 49 - 36 = 13 \ \mathrm{J}$$

Q: A skier of mass 70 kg descends a 50 m high hill. If 1000 J of energy is lost to friction, what is the skier's speed at the bottom?

$$v = \sqrt{2gh - \frac{2W_f}{m}} = \sqrt{2 \times 9.8 \times 50 - \frac{2 \times 1000}{70}} = \sqrt{980 - 28.57} = \sqrt{951.43} = 30.85 \ \mathrm{m/s}$$

Question 1

A 30 g marble is released from rest at the top of a ramp 62 cm above a table. The table is 95 cm above the floor. The marble leaves the ramp horizontally and lands 100 cm from the edge of the table. Assuming no energy is lost to friction, what would be the theoretical speed of the marble as it leaves the ramp?

Accepted Answer

$$v = \sqrt{2gh} = \sqrt{2 	imes 9.8 \ \mathrm{m/s^2$} 	imes 0.62 \ \mathrm{m}} = 3.49 \ \mathrm{m/s}$$

Question 2

If the marble in the previous question actually lands 100 cm from the table edge, but the calculated range (assuming no friction) is 120 cm, what can you conclude about the work done by friction?

Accepted Answer

Friction did negative work, removing mechanical energy from the system.

Question 3

A skier of mass $$m$$ starts from rest at a height $$h$$ and skis down a frictionless slope. What is the skier's speed at the bottom of the slope?

Accepted Answer

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

Question 4

A roller coaster car of mass $$m$$ must complete a vertical loop of radius $$R. $$What is the minimum speed at the top of the loop to avoid falling?

Accepted Answer

$$v_{	ext{min}} = \sqrt{gR}$$

Question 5

A 0.030 kg marble is released from a height of 62 cm above a table and lands 100 cm from the table edge. If the table is 95 cm above the floor, what is the total mechanical energy lost to friction if the theoretical speed is 3.49 m/s but the actual speed is 3.00 m/s?

Accepted Answer

$$\Delta E = \frac{1}{2} m (v_{	ext{theory}}^2 - v_{	ext{actual}}^2) = \frac{1}{2} 	imes 0.030 	imes (3.49^2$ - 3.00^2$) = 0.077 \ \mathrm{J}$$

Question 6

Which of the following statements about conservative and non-conservative forces is correct?

Accepted Answer

Friction is a non-conservative force because it dissipates mechanical energy as heat.

Question 7

A car of mass $$m$$ starts from rest at a height $$h$$ and enters a vertical loop of radius $$R. $$What is the minimum height $$h$$ required for the car to complete the loop without falling?

Accepted Answer

$$h = \frac{5}{2}R$$

Question 8

A skier descends a slope with friction. If the work done by friction is $$W_f$$, what is the skier's speed at the bottom if they start from rest at height $$h$$?

Accepted Answer

$$v = \sqrt{2gh - \frac{2W_f}{m}}$$

Question 9

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

Accepted Answer

The total mechanical energy (kinetic + potential) of a system remains constant if only conservative forces are acting.

Question 10

A marble rolls down two different tracks from the same height, one straight and one curved. Neglecting friction, which statement is true about the speed of the marble at the bottom?

Accepted Answer

The marble will have the same speed at the bottom for both tracks.

Question 11

A 0.5 kg object is dropped from a height of 10 m. If it lands with a speed of 12 m/s, how much energy was lost to air resistance?

Accepted Answer

$$\Delta E = mgh - \frac{1}{2}mv^2$ = 0.5 	imes 9.8 	imes 10 - 0.5 	imes 0.5 	imes 12^2$ = 49 - 36 = 13 \ \mathrm{J}$$

Question 12

A roller coaster car of mass $$m is at $$the top of a hill of height $$h. It $$descends and enters a loop of radius $$R. If $$friction is not negligible, which of the following must be true for the car to complete the loop?

Accepted Answer

The initial height $$h$$ must be greater than $$\frac{5}{2}R to $$compensate for energy lost to friction.

Question 13

Which of the following is an example of a non-conservative force?

Accepted Answer

Air resistance

Question 14

A skier of mass 70 kg descends a 50 m high hill. If 1000 J of energy is lost to friction, what is the skier's speed at the bottom?

Accepted Answer

$$v = \sqrt{2gh - \frac{2W_f}{m}} = \sqrt{2 	imes 9.8 	imes 50 - \frac{2 	imes 1000}{70}} = \sqrt{980 - 28.57} = \sqrt{951.43} = 30.85 \ \mathrm{m/s}$$

Question 15

A marble is launched horizontally from a table 95 cm above the floor and lands 100 cm from the edge. What is the time it takes to hit the floor?

Accepted Answer

$$t = \sqrt{\frac{2h}{g}} = \sqrt{\frac{2 	imes 0.95}{9.8}} = 0.44 \ \mathrm{s}$$

Question 16

A skier starts from rest at the top of a hill and descends a distance $$d$$ down the slope. If the coefficient of kinetic friction is $$\mu_k$$, what is the work done by friction?

Accepted Answer

$$W_f = -\mu_k mgd \cos 	heta$$

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