Physical Science Final Review: Kinetic and Potential Energy Calculations

Fundamental Principles and Equations of Energy

To perform the following calculations accurately, two primary forms of mechanical energy are considered: Kinetic Energy ($KE$) and Gravitational Potential Energy ($PE$). Kinetic energy is associated with an object's motion, while potential energy is associated with an object's position or height relative to a reference point.

• Kinetic Energy Formula: The energy of motion is calculated using the following equation:     $KE = \frac{1}{2}mv^{2}$     Where:

  • $KE$ is Kinetic Energy measured in Joules ($J$).

  • $m$ is the mass of the object measured in kilograms ($kg$).

  • $v$ is the velocity or speed of the object measured in meters per second ($m/s$).

• Potential Energy Formula: The energy of position is calculated using the following equation:     $PE = mgh$     Where:

  • $PE$ is Potential Energy measured in Joules ($J$).

  • $m$ is the mass of the object measured in kilograms ($kg$).

  • $g$ is the acceleration due to gravity, which is a constant approximately equal to $9.8\,m/s^{2}$.

  • $h$ is the height of the object above the reference surface measured in meters ($m$).

Problem-Solving Analysis

1. Kinetic Energy of a Motorcycle

• Step 1: Identify Given Information and Unknowns

  • Mass ($m$) = $900\,kg$

  • Velocity ($v$) = $25\,m/s$

  • Unknown: Kinetic Energy ($KE$)

• Step 2: Formula Selection

  • $KE = \frac{1}{2}mv^{2}$

• Step 3: Substitution

  • $KE = \frac{1}{2} \times 900\,kg \times (25\,m/s)^{2}$

• Step 4: Calculation

  • $KE = 450\,kg \times 625\,m^{2}/s^{2}$

• Step 5: Final Result

  • $KE = 281,250\,J$

2. Mass of a Hanging Sign

• Step 1: Identify Given Information and Unknowns

  • Potential Energy ($PE$) = $735\,J$

  • Height ($h$) = $5\,m$

  • Gravity ($g$) = $9.8\,m/s^{2}$

  • Unknown: Mass ($m$)

• Step 2: Formula Selection

  • $m = \frac{PE}{gh}$

• Step 3: Substitution

  • $m = \frac{735\,J}{(9.8\,m/s^{2})(5\,m)}$

• Step 4: Calculation

  • $m = \frac{735\,J}{49\,m^{2}/s^{2}}$

• Step 5: Final Result

  • $m = 15\,kg$

3. Kinetic Energy of a Skateboarder

• Step 1: Identify Given Information and Unknowns

  • Mass ($m$) = $65\,kg$

  • Velocity ($v$) = $8\,m/s$

  • Unknown: Kinetic Energy ($KE$)

• Step 2: Formula Selection

  • $KE = \frac{1}{2}mv^{2}$

• Step 3: Substitution

  • $KE = \frac{1}{2} \times 65\,kg \times (8\,m/s)^{2}$

• Step 4: Calculation

  • $KE = 32.5\,kg \times 64\,m^{2}/s^{2}$

• Step 5: Final Result

  • $KE = 2,080\,J$

4. Potential Energy of a Backpack

• Step 1: Identify Given Information and Unknowns

  • Mass ($m$) = $12\,kg$

  • Height ($h$) = $2.5\,m$

  • Gravity ($g$) = $9.8\,m/s^{2}$

  • Unknown: Potential Energy ($PE$)

• Step 2: Formula Selection

  • $PE = mgh$

• Step 3: Substitution

  • $PE = 12\,kg \times 9.8\,m/s^{2} \times 2.5\,m$

• Step 4: Calculation

  • $PE = 117.6\,N \times 2.5\,m$

• Step 5: Final Result

  • $PE = 294\,J$

5. Mass of a Diver

• Step 1: Identify Given Information and Unknowns

  • Height ($h$) = $10\,m$

  • Potential Energy ($PE$) = $5,880\,J$

  • Gravity ($g$) = $9.8\,m/s^{2}$

  • Unknown: Mass ($m$)

• Step 2: Formula Selection

  • $m = \frac{PE}{gh}$

• Step 3: Substitution

  • $m = \frac{5,880\,J}{(9.8\,m/s^{2})(10\,m)}$

• Step 4: Calculation

  • $m = \frac{5,880\,J}{98\,m^{2}/s^{2}}$

• Step 5: Final Result

  • $m = 60\,kg$

6. Kinetic Energy of a Car

• Step 1: Identify Given Information and Unknowns

  • Mass ($m$) = $1,500\,kg$

  • Velocity ($v$) = $20\,m/s$

  • Unknown: Kinetic Energy ($KE$)

• Step 2: Formula Selection

  • $KE = \frac{1}{2}mv^{2}$

• Step 3: Substitution

  • $KE = \frac{1}{2} \times 1,500\,kg \times (20\,m/s)^{2}$

• Step 4: Calculation

  • $KE = 750\,kg \times 400\,m^{2}/s^{2}$

• Step 5: Final Result

  • $KE = 300,000\,J$

7. Height of a Box

• Step 1: Identify Given Information and Unknowns

  • Mass ($m$) = $40\,kg$

  • Potential Energy ($PE$) = $1,960\,J$

  • Gravity ($g$) = $9.8\,m/s^{2}$

  • Unknown: Height ($h$)

• Step 2: Formula Selection

  • $h = \frac{PE}{mg}$

• Step 3: Substitution

  • $h = \frac{1,960\,J}{(40\,kg)(9.8\,m/s^{2})}$

• Step 4: Calculation

  • $h = \frac{1,960\,J}{392\,kg\cdot m/s^{2}}$

• Step 5: Final Result

  • $h = 5\,m$

8. Kinetic Energy of a Baseball

• Step 1: Identify Given Information and Unknowns

  • Mass ($m$) = $0.15\,kg$

  • Velocity ($v$) = $30\,m/s$

  • Unknown: Kinetic Energy ($KE$)

• Step 2: Formula Selection

  • $KE = \frac{1}{2}mv^{2}$

• Step 3: Substitution

  • $KE = \frac{1}{2} \times 0.15\,kg \times (30\,m/s)^{2}$

• Step 4: Calculation

  • $KE = 0.075\,kg \times 900\,m^{2}/s^{2}$

• Step 5: Final Result

  • $KE = 67.5\,J$

9. Potential Energy of a Rock

• Step 1: Identify Given Information and Unknowns

  • Mass ($m$) = $75\,kg$

  • Height ($h$) = $18\,m$

  • Gravity ($g$) = $9.8\,m/s^{2}$

  • Unknown: Potential Energy ($PE$)

• Step 2: Formula Selection

  • $PE = mgh$

• Step 3: Substitution

  • $PE = 75\,kg \times 9.8\,m/s^{2} \times 18\,m$

• Step 4: Calculation

  • $PE = 735\,N \times 18\,m$

• Step 5: Final Result

  • $PE = 13,230\,J$

10. Kinetic Energy of a Soccer Ball

• Step 1: Identify Given Information and Unknowns

  • Mass ($m$) = $2\,kg$

  • Velocity ($v$) = $12\,m/s$

  • Unknown: Kinetic Energy ($KE$)

• Step 2: Formula Selection

  • $KE = \frac{1}{2}mv^{2}$

• Step 3: Substitution

  • $KE = \frac{1}{2} \times 2\,kg \times (12\,m/s)^{2}$

• Step 4: Calculation

  • $KE = 1\,kg \times 144\,m^{2}/s^{2}$

• Step 5: Final Result

  • $KE = 144\,J$