The energy required to accelerate a car from 10 m/s to 20 m / s is how many times the energy required to accelerate the car from rest to 10 m /s

Options

A. Equal to the acceleration of the particle
B. 4 times
C. 2 times
D. 3 times

Answer

To solve the problem of how many times the energy required to accelerate a car from 10 m/s to 20 m/s is compared to the energy required to accelerate the car from rest to 10 m/s, we can use the concept of kinetic energy and the work-energy theorem.

### Step-by-Step Solution:

1. **Understand the Kinetic Energy Formula:**
The kinetic energy (KE) of an object is given by the formula:
\[ KE = \frac{1}{2} m v^2 \]
where \(m\) is the mass of the object and \(v\) is its velocity.

2. **Calculate the Change in Kinetic Energy from 10 m/s to 20 m/s:**
- Initial velocity (\(v_i\)) = 10 m/s
- Final velocity (\(v_f\)) = 20 m/s
- Change in kinetic energy (\(E_1\)):
\[ E_1 = KE_{final} - KE_{initial} = \frac{1}{2} m (20^2) - \frac{1}{2} m (10^2) \]
\[ E_1 = \frac{1}{2} m (400 - 100) = \frac{1}{2} m (300) = 150m \]

3. **Calculate the Change in Kinetic Energy from Rest to 10 m/s:**
- Initial velocity (\(v_i\)) = 0 m/s
- Final velocity (\(v_f\)) = 10 m/s
- Change in kinetic energy (\(E_2\)):
\[ E_2 = KE_{final} - KE_{initial} = \frac{1}{2} m (10^2) - \frac{1}{2} m (0^2) \]
\[ E_2 = \frac{1}{2} m (100) = 50m \]

4. **Determine the Ratio of Energies:**
Now, we need to find how many times \(E_1\) is compared to \(E_2\):
\[ \text{Ratio} = \frac{E_1}{E_2} = \frac{150m}{50m} \]
The mass \(m\) cancels out:
\[ \text{Ratio} = \frac{150}{50} = 3 \]

5. **Conclusion:**
The energy required to accelerate the car from 10 m/s to 20 m/s is **3 times** the energy required to accelerate the car from rest to 10 m/s.