Math solution on Kinetic Energy and Potential Energy

Kinetic Energy Problems

1. Problem: Calculate the kinetic energy of a 5 kg object moving at a velocity of 10 m/s.

   • Solution:
     $$𝐾𝐸=\frac{1}{2}𝑚{𝑣}^2=\frac{1}{2}\times 5\times 10^2=\frac{1}{2}\times 5\times 100=250\text{ J}$$

2. Problem: A car of mass 1000 kg is moving with a speed of 20 m/s. Find its kinetic energy.

   • Solution:
     $$𝐾𝐸=\frac{1}{2}𝑚{𝑣}^2=\frac{1}{2}\times 1000\times 20^2=\frac{1}{2}\times 1000\times 400=200,000\text{ J}$$

3. Problem: If a ball has a kinetic energy of 150 J and a mass of 2 kg, what is its velocity?

   • Solution:
     $$𝐾𝐸=\frac{1}{2}𝑚{𝑣}^2⟹150=\frac{1}{2}\times 2\times {𝑣}^2⟹150={𝑣}^2⟹𝑣=\sqrt{150}\approx 12.25\text{ m/s}$$

4. Problem: A 10 kg object is moving at 15 m/s. Calculate its kinetic energy.

   • Solution:
     $$𝐾𝐸=\frac{1}{2}𝑚{𝑣}^2=\frac{1}{2}\times 10\times 15^2=\frac{1}{2}\times 10\times 225=1125\text{ J}$$

5. Problem: What is the kinetic energy of a 0.5 kg ball traveling at 30 m/s?

   • Solution:
     $$𝐾𝐸=\frac{1}{2}𝑚{𝑣}^2=\frac{1}{2}\times 0.5\times 30^2=\frac{1}{2}\times 0.5\times 900=225\text{ J}$$

Potential Energy Problems

6. Problem: Calculate the gravitational potential energy of a 20 kg object raised to a height of 10 meters (g = 9.8 m/s²).

   • Solution:
     $$𝑃𝐸=𝑚𝑔ℎ=20\times 9.8\times 10=1960\text{ J}$$

7. Problem: A 50 kg object is lifted to a height of 5 meters. Find its potential energy.

   • Solution:
     $$𝑃𝐸=𝑚𝑔ℎ=50\times 9.8\times 5=2450\text{ J}$$

8. Problem: Determine the potential energy of a 2 kg object at a height of 8 meters.

   • Solution:
     $$𝑃𝐸=𝑚𝑔ℎ=2\times 9.8\times 8=156.8\text{ J}$$

9. Problem: A 10 kg object is at a height of 3 meters above the ground. What is its potential energy?

   • Solution:
     $$𝑃𝐸=𝑚𝑔ℎ=10\times 9.8\times 3=294\text{ J}$$

10. Problem: Find the potential energy of a 5 kg object placed 12 meters above the ground.

    • Solution:
      $$𝑃𝐸=𝑚𝑔ℎ=5\times 9.8\times 12=588\text{ J}$$

These problems cover basic calculations of kinetic and potential energy using their respective formulas:

• Kinetic Energy: $𝐾𝐸=\frac{1}{2}𝑚{𝑣}^2$
• Potential Energy: $𝑃𝐸=𝑚𝑔ℎ$

where $𝑚$ is mass, $𝑣$ is velocity, $𝑔$ is the acceleration due to gravity (approximately 9.8 m/s²), and $ℎ$ is height.