A boy is pushing a box on horizontal floor from a position of rest to rest , while moving along a straight line. Consider the three phases of motion. The floor is rough with a small friction coefficient .
(i) Initially a constant hard push on the box to get it moving and attain a maximum velocity .
(ii) Mild push to keep the box moving with constant velocity
(iii) To pull back the box the to bring it to stop with the same retardation.
Which of the following graph is CORRECT ?

Options

A.
B.
C.
D.

Answer

To solve the problem, we need to analyze the three phases of motion of the box as the boy pushes it along a rough horizontal floor. We will consider the forces acting on the box and how they change during each phase.

### Step-by-Step Solution:

1. **Phase 1: Initial Push to Get the Box Moving**
- The boy applies a constant hard push to overcome static friction and get the box moving.
- Initially, the applied force must be greater than the static friction force to initiate motion.
- As the box starts moving, the friction changes from static to kinetic.
- The applied force increases until it reaches a maximum value (limiting friction), after which it can decrease slightly to maintain motion.
- **Graph Representation**: The applied force will start high and then decrease slightly as the box moves with maximum velocity.

2. **Phase 2: Mild Push to Maintain Constant Velocity**
- Once the box is moving, the boy applies a mild push to keep the box moving at a constant velocity.
- The applied force in this phase is equal to the kinetic friction force acting on the box.
- Since the velocity is constant, the net force is zero, meaning the applied force is balanced by the frictional force.
- **Graph Representation**: The applied force remains constant and lower than the maximum force applied in Phase 1.

3. **Phase 3: Pulling Back to Bring the Box to a Stop**
- To stop the box, the boy applies a force in the opposite direction.
- This force must be greater than the kinetic friction force acting on the box to bring it to rest.
- The applied force will start at a certain value and decrease as the box slows down, eventually reaching zero when the box comes to rest.
- **Graph Representation**: The applied force will start from a positive value and decrease to zero as the box comes to rest.

### Conclusion:
The correct graph for the applied force versus time will show:
- An initial increase in force during the first phase, followed by a decrease as the box reaches maximum velocity.
- A constant force during the second phase.
- A decrease in force during the third phase as the box comes to a stop.