A body is acted on by a force given by F = (10+2t) N. The impulse received by the body during the first four seconds is

Options

A. 40 N s
B. 56 N s
C. 72 N s
D. 32 N s

Answer

To solve the problem of finding the impulse received by a body acted upon by a force \( F = (10 + 2t) \) N during the first four seconds, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Impulse**:
Impulse is defined as the change in momentum of an object, which can also be calculated as the integral of force over time. Mathematically, this is expressed as:
\[ \text{Impulse} = \int F \, dt \]

2. **Setting Up the Integral**:
Given the force \( F = 10 + 2t \), we need to calculate the impulse from \( t = 0 \) to \( t = 4 \) seconds. Thus, we set up the integral:
\[ \text{Impulse} = \int_{0}^{4} (10 + 2t) \, dt \]

3. **Calculating the Integral**:
To solve the integral, we can break it down:
\[ \int (10 + 2t) \, dt = \int 10 \, dt + \int 2t \, dt \]
Calculating each part:
- The integral of \( 10 \) with respect to \( t \) is:
\[ 10t \]
- The integral of \( 2t \) with respect to \( t \) is:
\[ t^2 \]
Therefore, we have:
\[ \int (10 + 2t) \, dt = 10t + t^2 \]

4. **Evaluating the Integral from 0 to 4**:
Now we evaluate the expression \( 10t + t^2 \) at the limits \( t = 0 \) and \( t = 4 \):
\[ \text{Impulse} = \left[ 10(4) + (4)^2 \right] - \left[ 10(0) + (0)^2 \right] \]
Calculating this gives:
\[ \text{Impulse} = (40 + 16) - (0) = 56 \, \text{N·s} \]

5. **Final Result**:
Thus, the impulse received by the body during the first four seconds is:
\[ \text{Impulse} = 56 \, \text{N·s} \]

### Summary:
The impulse received by the body during the first four seconds is \( 56 \, \text{N·s} \).