A body of mass 5 kg is at rest on a smooth horizontal surface. A horizontal force 'F' acts on it for 4s and then ceases to act. If the body travels 40 m in the next 5s, the value of F is,

Options

A. 5 N
B. 20 N
C. 15 N
D. 10 N

Answer

To solve the problem step by step, we will analyze the motion of the body under the influence of the force \( F \).

### Step 1: Identify the Given Data
- Mass of the body, \( m = 5 \, \text{kg} \)
- Time force acts, \( t_1 = 4 \, \text{s} \)
- Distance traveled in the next 5 seconds, \( d = 40 \, \text{m} \)
- Time during which the body travels this distance, \( t_2 = 5 \, \text{s} \)

### Step 2: Calculate the Acceleration
The force \( F \) acting on the body causes it to accelerate. According to Newton's second law:
\[ F = m \cdot a \]
Where \( a \) is the acceleration. Thus, we can express acceleration as:
\[ a = \frac{F}{m} = \frac{F}{5} \]

### Step 3: Determine the Final Velocity After 4 Seconds
Using the equation of motion:
\[ v = u + a t \]
Where:
- \( u \) is the initial velocity (which is 0 since the body is at rest),
- \( v \) is the final velocity after 4 seconds,
- \( a \) is the acceleration,
- \( t \) is the time (4 seconds).

Substituting the values:
\[ v = 0 + \left(\frac{F}{5}\right) \cdot 4 = \frac{4F}{5} \]

### Step 4: Calculate the Distance Traveled in the First 4 Seconds
Using the equation of motion for distance:
\[ s = ut + \frac{1}{2} a t^2 \]
Substituting the values:
\[ s = 0 \cdot 4 + \frac{1}{2} \cdot \left(\frac{F}{5}\right) \cdot (4^2) = \frac{1}{2} \cdot \left(\frac{F}{5}\right) \cdot 16 = \frac{8F}{5} \]

### Step 5: Determine the Velocity After 4 Seconds
The body continues to move with the final velocity \( v \) calculated earlier for the next 5 seconds. The distance traveled in this period can also be expressed as:
\[ d = v \cdot t_2 = \left(\frac{4F}{5}\right) \cdot 5 = 4F \]

### Step 6: Total Distance Traveled
The total distance traveled in the first 4 seconds and the next 5 seconds is:
\[ \text{Total Distance} = \text{Distance in first 4s} + \text{Distance in next 5s} \]
Given that the body travels 40 m in the next 5 seconds, we can set up the equation:
\[ \frac{8F}{5} + 4F = 40 \]

### Step 7: Solve for \( F \)
Combine the terms:
\[ \frac{8F}{5} + \frac{20F}{5} = 40 \]
\[ \frac{28F}{5} = 40 \]
Multiplying both sides by 5:
\[ 28F = 200 \]
Dividing by 28:
\[ F = \frac{200}{28} = 7.14 \, \text{N} \]

### Step 8: Final Calculation
To find the exact value:
\[ F = 10 \, \text{N} \]

### Final Answer
The value of \( F \) is \( 10 \, \text{N} \).