A body of mass 10 kg is being acted upon by a force `3t^2` and an opposing constant force of 32 N. The initial speed is `10 ms^-1.`The velocity of body after 5 s is

Options

A. `14.5 ms^-1`
B. `6.5 ms^-1`
C. `3.5 ms^-1`
D. `4.5 ms^-1`

Explanation

To solve the problem step by step, we will follow the physics principles of force, acceleration, and integration to find the velocity of the body after 5 seconds.

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Body:**
- The body has a mass \( m = 10 \, \text{kg} \).
- The force acting on the body is \( F(t) = 3t^2 \) (in Newtons).
- There is an opposing constant force of \( 32 \, \text{N} \).

2. **Calculate the Net Force:**
- The net force \( F_{\text{net}} \) acting on the body is given by:
\[ F_{\text{net}} = F(t) - \text{Opposing Force} = 3t^2 - 32 \]

3. **Determine the Acceleration:**
- Using Newton's second law, \( F = ma \), we can find the acceleration \( a(t) \):
\[ a(t) = \frac{F_{\text{net}}}{m} = \frac{3t^2 - 32}{10} \]
- Simplifying this gives:
\[ a(t) = 0.3t^2 - 3.2 \]

4. **Relate Acceleration to Velocity:**
- We know that acceleration is the derivative of velocity with respect to time:
\[ a(t) = \frac{dv}{dt} \]
- Therefore, we can write:
\[ \frac{dv}{dt} = 0.3t^2 - 3.2 \]

5. **Integrate to Find Velocity:**
- To find the velocity, we integrate the acceleration with respect to time:
\[ v(t) = \int (0.3t^2 - 3.2) \, dt \]
- This gives:
\[ v(t) = 0.1t^3 - 3.2t + C \]
- Where \( C \) is the constant of integration.

6. **Determine the Constant of Integration:**
- We know the initial velocity \( v(0) = 10 \, \text{m/s} \):
\[ v(0) = 0.1(0)^3 - 3.2(0) + C = 10 \]
- Therefore, \( C = 10 \).

7. **Final Velocity Equation:**
- Substituting \( C \) back into the velocity equation:
\[ v(t) = 0.1t^3 - 3.2t + 10 \]

8. **Calculate the Velocity at \( t = 5 \, \text{s} \):**
- Now, substitute \( t = 5 \) into the velocity equation:
\[ v(5) = 0.1(5)^3 - 3.2(5) + 10 \]
- Calculating this:
\[ v(5) = 0.1(125) - 16 + 10 = 12.5 - 16 + 10 = 6.5 \, \text{m/s} \]

### Final Answer:
The velocity of the body after 5 seconds is \( 6.5 \, \text{m/s} \).

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