Q: A 60 kg roller coaster starts from rest at a height of 30 m and descends to a height of 5 m. Neglecting friction, what is its speed at the lower height? (Use $g = 9.8\ \mathrm{m/s^2}$)

$v = \sqrt{2g(h_i - h_f)} = \sqrt{2 \times 9.8 \times (30 - 5)} = 22.1\ \mathrm{m/s}$

Q: A 5 kg object is dropped from a height of 10 m. What is its kinetic energy just before hitting the ground? (Neglect air resistance, $g = 9.8\ \mathrm{m/s^2}$)

$K = mgh = 5 \times 9.8 \times 10 = 490\ \mathrm{J}$

Q: A 2 kg mass is attached to a spring ($k = 200\ \mathrm{N/m}$) and pulled 0.1 m from equilibrium. What is the total mechanical energy if released from rest at this position?

$E_{mech} = \frac{1}{2}kx^2 = \frac{1}{2} \times 200 \times (0.1)^2 = 1\ \mathrm{J}$

Q: A 1000 kg roller coaster car starts from rest at a height of 40 m. If it reaches the bottom at 5 m with a speed of 25 m/s, how much energy was lost to friction? ($g = 9.8\ \mathrm{m/s^2}$)

$E_{lost} = mgh_i - (mgh_f + \frac{1}{2}mv^2) = 1000 \times 9.8 \times 40 - [1000 \times 9.8 \times 5 + 0.5 \times 1000 \times 25^2] = 392,000 - (49,000 + 312,500) = 30,500\ \mathrm{J}$

Question 1

Which of the following best describes the law of conservation of energy as presented in the study materials?

Accepted Answer

The total energy of an isolated system remains constant over time, though it may change forms.

Question 2

A 60 kg roller coaster starts from rest at a height of 30 m and descends to a height of 5 m. Neglecting friction, what is its speed at the lower height? (Use $g = 9.8\ \mathrm{m/s^2}$)

Accepted Answer

$v = \sqrt{2g(h_i - h_f)} = \sqrt{2 	imes 9.8 	imes (30 - 5)} = 22.1\ \mathrm{m/s}$

Question 3

Which equation correctly expresses the total mechanical energy $E_{mech}$ of a system as defined in the materials?

Accepted Answer

$E_{mech} = K + U$

Question 4

A mass $m$ is attached to an ideal spring with spring constant $k$ and is displaced by $x$ from equilibrium. What is the elastic potential energy stored in the spring?

Accepted Answer

$U_s = \frac{1}{2}kx^2$

Question 5

If a system loses mechanical energy due to friction, which of the following statements is true according to the study materials?

Accepted Answer

The lost mechanical energy is transformed into other forms, such as thermal energy, but the total energy of the system plus environment is conserved.

Question 6

Which of the following is NOT a type of energy explicitly listed in the study materials?

Accepted Answer

Magnetic energy

Question 7

A 5 kg object is dropped from a height of 10 m. What is its kinetic energy just before hitting the ground? (Neglect air resistance, $g = 9.8\ \mathrm{m/s^2}$)

Accepted Answer

$K = mgh = 5 	imes 9.8 	imes 10 = 490\ \mathrm{J}$

Question 8

Which of the following best explains why the final speed of a roller coaster at the bottom of a hill does NOT depend on its mass (neglecting friction)?

Accepted Answer

Both kinetic and potential energy are proportional to mass, so mass cancels out when applying conservation of energy.

Question 9

Which formula represents the change in potential energy between two positions $1$ and $2$?

Accepted Answer

$\Delta U = U_2 - U_1$

Question 10

A 2 kg mass is attached to a spring ($k = 200\ \mathrm{N/m}$) and pulled 0.1 m from equilibrium. What is the total mechanical energy if released from rest at this position?

Accepted Answer

$E_{mech} = \frac{1}{2}kx^2 = \frac{1}{2} 	imes 200 	imes (0.1)^2 = 1\ \mathrm{J}$

Question 11

Which of the following is a correct statement about potential energy as discussed in the materials?

Accepted Answer

Potential energy is a relative quantity; only changes in potential energy between two positions are physically meaningful.

Question 12

A 1000 kg roller coaster car starts from rest at a height of 40 m. If it reaches the bottom at 5 m with a speed of 25 m/s, how much energy was lost to friction? ($g = 9.8\ \mathrm{m/s^2}$)

Accepted Answer

$E_{lost} = mgh_i - (mgh_f + \frac{1}{2}mv^2) = 1000 	imes 9.8 	imes 40 - [1000 	imes 9.8 	imes 5 + 0.5 	imes 1000 	imes 25^2] = 392,000 - (49,000 + 312,500) = 30,500\ \mathrm{J}$

Question 13

Which of the following is the correct expression for the kinetic energy of a particle of mass $m$ and speed $v$?

Accepted Answer

$K = \frac{1}{2}mv^2$

Question 14

If the total mechanical energy of a system is constant, which of the following must be true?

Accepted Answer

The sum of kinetic and potential energy remains the same at all times.

Question 15

A block slides down a frictionless incline, converting potential energy to kinetic energy. Which equation best describes this process?

Accepted Answer

$K_i + U_i = K_f + U_f$

Question 16

Which of the following is a correct statement about the relationship between equations of motion and conservation laws?

Accepted Answer

Equations of motion describe how things change with time, while conservation laws describe physical quantities that remain constant in a given physical scenario.

Question 17

A 0.5 kg mass is attached to a spring ($k = 100\ \mathrm{N/m}$) and oscillates without friction. If released from rest at $x = 0.2\ \mathrm{m}$, what is its speed as it passes through equilibrium ($x = 0$)?

Accepted Answer

$v = \sqrt{\frac{k}{m}}x_0 = \sqrt{\frac{100}{0.5}} 	imes 0.2 = 2.83\ \mathrm{m/s}$

Question 18

Which of the following is a correct application of the conservation of energy principle to a real-world scenario?

Accepted Answer

When a car brakes to a stop, its kinetic energy is transformed into thermal energy in the brakes and road.

Conservation of Energy: Principles, Types, and Applications

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