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 analyze the forces acting on the body, calculate the acceleration, and then find the final velocity after 5 seconds.

### Step 1: Identify the forces acting on the body
The body is subjected to two forces:
1. A time-dependent force: \( F(t) = 3t^2 \) N
2. An opposing constant force: \( F_{\text{opposing}} = 32 \) N

### Step 2: Calculate the net force acting on the body
The net force \( F_{\text{net}} \) acting on the body can be expressed as:
\[ F_{\text{net}} = F(t) - F_{\text{opposing}} = 3t^2 - 32 \]

### Step 3: Calculate the acceleration of the body
Using Newton's second law, the acceleration \( a \) can be calculated as:
\[ a = \frac{F_{\text{net}}}{m} \]
where \( m = 10 \) kg (mass of the body). Therefore,
\[ a = \frac{3t^2 - 32}{10} = 0.3t^2 - 3.2 \text{ m/s}^2 \]

### Step 4: Set up the equation for velocity
We know that the change in velocity can be calculated using the integral of acceleration over time. The initial velocity \( u = 10 \) m/s. We need to find the final velocity \( v \) after \( t = 5 \) seconds. The equation can be set up as:
\[ v - u = \int_{0}^{5} a \, dt \]
Substituting the expression for acceleration:
\[ v - 10 = \int_{0}^{5} (0.3t^2 - 3.2) \, dt \]

### Step 5: Calculate the integral
Now we compute the integral:
\[ \int (0.3t^2 - 3.2) \, dt = 0.1t^3 - 3.2t \]
Evaluating this from \( t = 0 \) to \( t = 5 \):
\[ \left[ 0.1(5)^3 - 3.2(5) \right] - \left[ 0.1(0)^3 - 3.2(0) \right] \]
Calculating:
\[ = 0.1(125) - 3.2(5) = 12.5 - 16 = -3.5 \]

### Step 6: Solve for final velocity
Now substituting back into the equation for velocity:
\[ v - 10 = -3.5 \]
\[ v = 10 - 3.5 = 6.5 \text{ m/s} \]

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

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