A body of mass 4 kg is accelerated up by a constant force, travels a distance of 5 m in the first second and a distance of 2m in the third second. The force acting on the body is

Options

A. 2 N
B. 4 N
C. 6 N
D. 8 N

Answer

To solve the problem, we need to determine the force acting on a body of mass 4 kg that travels certain distances in specified time intervals. We will use the equations of motion to derive the necessary values step by step.

### Step 1: Analyze the given information
We know:
- Mass of the body, \( m = 4 \, \text{kg} \)
- Distance traveled in the first second, \( S_1 = 5 \, \text{m} \)
- Distance traveled in the third second, \( S_3 = 2 \, \text{m} \)

### Step 2: Use the equation of motion for the first second
The equation of motion for distance traveled in the first second is given by:
\[ S_1 = u + \frac{1}{2} a (t^2) \]
For \( t = 1 \) second, we have:
\[ 5 = u + \frac{1}{2} a (1^2) \]
This simplifies to:
\[ 5 = u + \frac{1}{2} a \quad \text{(Equation 1)} \]

### Step 3: Use the equation of motion for the second second
The total distance traveled in the first two seconds is:
\[ S_2 + S_1 = u(2) + \frac{1}{2} a (2^2) \]
Let \( S_2 \) be the distance traveled in the second second. We can express \( S_2 \) as:
\[ S_2 = S_1 + S_3 - S_1 = S_3 = 2 \, \text{m} \]
Thus, the equation becomes:
\[ 5 + 2 = 2u + 2a \]
This simplifies to:
\[ 7 = 2u + 2a \quad \text{(Equation 2)} \]

### Step 4: Use the equation of motion for the third second
The total distance traveled in three seconds is:
\[ S_1 + S_2 + S_3 = u(3) + \frac{1}{2} a (3^2) \]
Substituting the known values:
\[ 5 + 2 + 2 = 3u + \frac{9}{2} a \]
This simplifies to:
\[ 9 = 3u + \frac{9}{2} a \quad \text{(Equation 3)} \]

### Step 5: Solve the equations
Now we have three equations:
1. \( 5 = u + \frac{1}{2} a \)
2. \( 7 = 2u + 2a \)
3. \( 9 = 3u + \frac{9}{2} a \)

From Equation 1, we can express \( u \):
\[ u = 5 - \frac{1}{2} a \]

Substituting \( u \) into Equation 2:
\[ 7 = 2(5 - \frac{1}{2} a) + 2a \]
This simplifies to:
\[ 7 = 10 - a + 2a \]
\[ 7 = 10 + a \implies a = 7 - 10 = -3 \, \text{m/s}^2 \]

### Step 6: Calculate the force
Using Newton's second law:
\[ F = m \cdot a \]
Substituting the values:
\[ F = 4 \cdot (-3) = -12 \, \text{N} \]
Since the force is negative, it indicates that the force is acting in the opposite direction to the motion.

### Final Answer
The force acting on the body is \( -12 \, \text{N} \).