The ratio of energy required to accelerated a car from rest to `20ms^(-1)` to the energy needed to acceelerate from `20ms^(-1)` to `40ms^(-1)` is

Options

A. `1:1`
B. `1:3`
C. `1:2`
D. `1:4`

Answer

To solve the problem of finding the ratio of energy required to accelerate a car from rest to \(20 \, \text{m/s}\) and from \(20 \, \text{m/s}\) to \(40 \, \text{m/s}\), we will use the concept of kinetic energy. The work done in accelerating the car is equal to the change in kinetic energy.

### Step-by-Step Solution:

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

2. **Calculate the Energy Required to Accelerate from Rest to \(20 \, \text{m/s}\)**:
- Initial velocity \(u_1 = 0 \, \text{m/s}\) (at rest)
- Final velocity \(v_1 = 20 \, \text{m/s}\)
- Change in kinetic energy (\(\Delta KE_1\)):
\[ \Delta KE_1 = KE_{final} - KE_{initial} = \frac{1}{2} m (v_1^2) - \frac{1}{2} m (u_1^2) \]
\[ \Delta KE_1 = \frac{1}{2} m (20^2) - \frac{1}{2} m (0^2) = \frac{1}{2} m (400) = 200m \]

3. **Calculate the Energy Required to Accelerate from \(20 \, \text{m/s}\) to \(40 \, \text{m/s}\)**:
- Initial velocity \(u_2 = 20 \, \text{m/s}\)
- Final velocity \(v_2 = 40 \, \text{m/s}\)
- Change in kinetic energy (\(\Delta KE_2\)):
\[ \Delta KE_2 = KE_{final} - KE_{initial} = \frac{1}{2} m (v_2^2) - \frac{1}{2} m (u_2^2) \]
\[ \Delta KE_2 = \frac{1}{2} m (40^2) - \frac{1}{2} m (20^2) = \frac{1}{2} m (1600) - \frac{1}{2} m (400) \]
\[ \Delta KE_2 = \frac{1}{2} m (1600 - 400) = \frac{1}{2} m (1200) = 600m \]

4. **Finding the Ratio of Energies**:
Now we can find the ratio of the energy required for the two cases:
\[ \text{Ratio} = \frac{\Delta KE_1}{\Delta KE_2} = \frac{200m}{600m} = \frac{200}{600} = \frac{1}{3} \]

5. **Final Answer**:
The ratio of the energy required to accelerate from rest to \(20 \, \text{m/s}\) to the energy needed to accelerate from \(20 \, \text{m/s}\) to \(40 \, \text{m/s}\) is:
\[ 1 : 3 \]