Displacement x ( in meters ) , of a body of mass 1 kg as a function of time t, on a horizontal smooth surface , is given as `x = 2 t^2` Then work done in the first one second by the external force is

Options

A. 2 J
B. 4 J
C. 8 J
D. 16 J

Answer

To solve the problem, we need to find the work done by the external force on a body of mass 1 kg, given its displacement as a function of time \( x = 2t^2 \).

### Step-by-Step Solution:

1. **Identify the Displacement Function**:
The displacement \( x \) is given by:
\[ x = 2t^2 \]

2. **Differentiate to Find Velocity**:
To find the velocity \( v \), we differentiate \( x \) with respect to time \( t \):
\[ v = \frac{dx}{dt} = \frac{d(2t^2)}{dt} = 4t \]

3. **Differentiate to Find Acceleration**:
Next, we differentiate the velocity \( v \) with respect to time \( t \) to find the acceleration \( a \):
\[ a = \frac{dv}{dt} = \frac{d(4t)}{dt} = 4 \]

4. **Calculate the Force**:
Using Newton's second law, \( F = ma \), where \( m = 1 \, \text{kg} \) and \( a = 4 \, \text{m/s}^2 \):
\[ F = 1 \times 4 = 4 \, \text{N} \]

5. **Calculate the Work Done**:
Work done \( W \) is given by the integral of force over displacement. Since we know the force is constant, we can use the formula:
\[ W = F \cdot d \]
where \( d \) is the displacement in the first second. We first need to find the displacement at \( t = 1 \, \text{s} \):
\[ x(1) = 2(1^2) = 2 \, \text{m} \]
Therefore, the work done in the first second is:
\[ W = F \cdot d = 4 \, \text{N} \cdot 2 \, \text{m} = 8 \, \text{J} \]

### Final Answer:
The work done in the first one second by the external force is \( 8 \, \text{J} \).

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