A body starts from rest and is acted on by a constant force. The ratio of kinetic energy gained by it in the first five seconds to that gained in the next five seconds is

Options

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

Explanation

To solve the problem, we need to find the ratio of the kinetic energy gained by a body in the first five seconds to that gained in the next five seconds when acted upon by a constant force.

### Step-by-Step Solution:

1. **Understanding the Motion**:
The body starts from rest and is acted upon by a constant force. This means the acceleration (a) is constant.

2. **Kinematic Equations**:
Since the body starts from rest, the initial velocity (u) is 0. The final velocity (v) at any time t can be given by the equation:
\[ v = u + at = 0 + at = at \]

3. **Kinetic Energy Calculation**:
The kinetic energy (KE) of the body at any time t is given by:
\[ KE = \frac{1}{2} mv^2 \]
Substituting for v, we have:
\[ KE = \frac{1}{2} m (at)^2 = \frac{1}{2} m a^2 t^2 \]

4. **Kinetic Energy from 0 to 5 seconds**:
For the first 5 seconds (t = 5):
\[ KE_{0 \text{ to } 5} = \frac{1}{2} m a^2 (5)^2 = \frac{1}{2} m a^2 \cdot 25 = \frac{25}{2} m a^2 \]

5. **Kinetic Energy from 0 to 10 seconds**:
For the first 10 seconds (t = 10):
\[ KE_{0 \text{ to } 10} = \frac{1}{2} m a^2 (10)^2 = \frac{1}{2} m a^2 \cdot 100 = 50 m a^2 \]

6. **Kinetic Energy from 5 to 10 seconds**:
The kinetic energy gained from 5 to 10 seconds is the difference between the kinetic energy at 10 seconds and the kinetic energy at 5 seconds:
\[ KE_{5 \text{ to } 10} = KE_{0 \text{ to } 10} - KE_{0 \text{ to } 5} = 50 m a^2 - \frac{25}{2} m a^2 \]
To perform the subtraction:
\[ KE_{5 \text{ to } 10} = 50 m a^2 - 12.5 m a^2 = 37.5 m a^2 = \frac{75}{2} m a^2 \]

7. **Finding the Ratio**:
Now, we can find the ratio of the kinetic energy gained in the first 5 seconds to that gained in the next 5 seconds:
\[ \text{Ratio} = \frac{KE_{0 \text{ to } 5}}{KE_{5 \text{ to } 10}} = \frac{\frac{25}{2} m a^2}{\frac{75}{2} m a^2} = \frac{25}{75} = \frac{1}{3} \]

### Final Answer:
The ratio of kinetic energy gained by the body in the first five seconds to that gained in the next five seconds is \( \frac{1}{3} \).