Q: A 4 kg object is launched vertically upward from the ground with an initial speed of $$40 \ \mathrm{m/s}. $$What is the maximum height it reaches? (Ignore air resistance.)

$$h = \frac{v_0$^2}{2g} = \frac{40^2$}{2 \times 9.8} = 81.6 \ \mathrm{m}$$

Q: A 6 kg object is thrown downward from a height of $$20 \ \mathrm{m}$$ and reaches the ground with a speed of $$30 \ \mathrm{m/s}. $$What was the initial speed with which it was thrown?

$$v_0$$ = \sqrt{$$v^2 - 2g$$\Delta y} = \sqrt{$$30^2 - 2$$ \times 9.8 \times 20} = 10 \ \mathrm{m/s}$$

Q: A block slides along a flat surface at $$30 \ \mathrm{m/s}$$ and enters a rough patch with a coefficient of friction $$\mu = 0.6. $$How far does it travel before stopping? (Assume $$g = 9.8 \ \mathrm{m/s^2$}.)$$

$$d = \frac{v^2$}{2 \mu g} = \frac{30^2$}{2 \times 0.6 \times 9.8} = 76.5 \ \mathrm{m}$$

Q: A 2 kg object is dropped from $$80 \ \mathrm{m}$$ and reaches the ground with $$30 \ \mathrm{m/s}. $$What is the work done by air resistance?

$$W_{air} = \Delta K + \Delta U = (0.5 \times 2 \times 30^2$ - 0) + (0 - 2 \times 9.8 \times 80) = 900 - 1568 = -668 \ \mathrm{J}$$

Q: A 4 kg block moving at $$20 \ \mathrm{m/s}$$ strikes a massless spring ($$k = 600 \ \mathrm{N/m}) on a $$frictionless surface. What is the maximum compression of the spring?

$$\Delta x = \sqrt{\frac{mv^2$}{k}} = \sqrt{\frac{4 \times 400}{600}} = 1.63 \ \mathrm{m}$$

Question 1

A 2 kg ball is dropped from the top of a 100 m building. What is the total mechanical energy of the ball at the top of the building? (Assume $$g = 9.8 \ \mathrm{m/s^2$}$$ and take the ground as $$y=0.)$$

Accepted Answer

$$ME = K + U = 0 + (2)(9.8)(100) = 1960 \ \mathrm{J}$$

Question 2

Which of the following statements is TRUE regarding the conservation of mechanical energy in an isolated system?

Accepted Answer

Mechanical energy is conserved if only internal (conservative) forces do work.

Question 3

A 4 kg object is launched vertically upward from the ground with an initial speed of $$40 \ \mathrm{m/s}. $$What is the maximum height it reaches? (Ignore air resistance.)

Accepted Answer

$$h = \frac{v_0$^2}{2g} = \frac{40^2$}{2 	imes 9.8} = 81.6 \ \mathrm{m}$$

Question 4

A 6 kg object is thrown downward from a height of $$20 \ \mathrm{m}$$ and reaches the ground with a speed of $$30 \ \mathrm{m/s}. $$What was the initial speed with which it was thrown?

Accepted Answer

$$v_0$$ = \sqrt{$$v^2 - 2g$$\Delta y} = \sqrt{$$30^2 - 2$$ 	imes 9.8 	imes 20} = 10 \ \mathrm{m/s}$$

Question 5

Which of the following is a non-conservative force?

Accepted Answer

Friction

Question 6

A 0.5 kg hockey puck is moving at $$4 \ \mathrm{m/s} on $$smooth ice. You push it with a constant $$200 \ \mathrm{N}$$ force through $$0.3 \ \mathrm{m}. $$What is its final speed?

Accepted Answer

$$v_f = \sqrt{v_i^2$ + \frac{2F d}{m}} = \sqrt{4^2$ + \frac{2 	imes 200 	imes 0.3}{0.5}} = 15.62 \ \mathrm{m/s}$$

Question 7

A block slides along a flat surface at $$30 \ \mathrm{m/s}$$ and enters a rough patch with a coefficient of friction $$\mu = 0.6. $$How far does it travel before stopping? (Assume $$g = 9.8 \ \mathrm{m/s^2$}.)$$

Accepted Answer

$$d = \frac{v^2$}{2 \mu g} = \frac{30^2$}{2 	imes 0.6 	imes 9.8} = 76.5 \ \mathrm{m}$$

Question 8

A 2 kg object is dropped from $$80 \ \mathrm{m}$$ and reaches the ground with $$30 \ \mathrm{m/s}. $$What is the work done by air resistance?

Accepted Answer

$$W_{air} = \Delta K + \Delta U = (0.5 	imes 2 	imes 30^2$ - 0) + (0 - 2 	imes 9.8 	imes 80) = 900 - 1568 = -668 \ \mathrm{J}$$

Question 9

Which of the following best describes a stable equilibrium point on a potential energy graph $$U(x)$$?

Accepted Answer

A local minimum where a small displacement results in a restoring force back to equilibrium.

Question 10

A 4 kg block moving at $$20 \ \mathrm{m/s}$$ strikes a massless spring ($$k = 600 \ \mathrm{N/m}) on a $$frictionless surface. What is the maximum compression of the spring?

Accepted Answer

$$\Delta x = \sqrt{\frac{mv^2$}{k}} = \sqrt{\frac{4 	imes 400}{600}} = 1.63 \ \mathrm{m}$$

Question 11

A particle moves according to the potential energy function $$U(x). At $$which point is the force on the particle zero?

Accepted Answer

Where the slope of $$U(x) is $$zero (horizontal tangent).

Question 12

A 3 kg block hangs 2 m above the floor, connected to a 4 kg block on a rough table ($$\mu_k = 0.5). If $$released from rest, what is the speed of the 3 kg block just before hitting the floor? (Assume $$g = 9.8 \ \mathrm{m/s^2$}.)$$

Accepted Answer

$$v = \sqrt{\frac{2(m_1$ g h - \mu_k m_2$ g h)}{m_1$ + m_2$}} = \sqrt{\frac{2(3 	imes 9.8 	imes 2 - 0.5 	imes 4 	imes 9.8 	imes 2)}{7}} = 3.33 \ \mathrm{m/s}$$

Question 13

Which of the following is TRUE about the work done by non-conservative forces ($$W_{NC}) in $$the conservation of energy equation?

Accepted Answer

$$W_{NC}$$ accounts for energy added to or removed from the system by forces like friction or applied forces.

Question 14

A 30 kg block hangs above the floor, connected to a 20 kg block on a smooth incline at $$53^\circ. If $$released from rest, what is the speed of the system when the 30 kg block has moved 1 m? (Assume $$g = 9.8 \ \mathrm{m/s^2$}.)$$

Accepted Answer

$$v = \sqrt{\frac{2(m_1$ g h - m_2$ g h \sin 	heta)}{m_1$ + m_2$}} = \sqrt{\frac{2(30 	imes 9.8 	imes 1 - 20 	imes 9.8 	imes 1 	imes \sin 53^\circ)}{50}} = 1.98 \ \mathrm{m/s}$$

Question 15

Which of the following is TRUE about potential energy graphs $$U(x)$$?

Accepted Answer

The kinetic energy at any point is the difference between the total mechanical energy and $$U(x).$$

Question 16

A 4 kg block moving at $$20 \ \mathrm{m/s}$$ strikes a spring ($$k = 600 \ \mathrm{N/m}) on a $$surface with friction ($$\mu = 0.5). $$What is the maximum compression of the spring?

Accepted Answer

$$\Delta x = \sqrt{\frac{mv^2$}{k + 2\mu mg}} = \sqrt{\frac{4 	imes 400}{600 + 2 	imes 0.5 	imes 4 	imes 9.8}} = 1.36 \ \mathrm{m}$$

Conservation of Mechanical Energy and Potential Energy in Physics

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