When the angle of inclination of on inclined plane is `theta`, an object slides down with uniform velocity. If the same object is pushed up with an initial velocity u on the same inclined plane, it goes up the plane and stops at a certain distance on the plane. There after the body.

Options

A. Slides down the inclined plane and reaches the ground. With velocity "u"
B. Slides down the inclined plane and reaches the ground with velocity less than "u"
C. Slides down the inclined plane and reaches the ground with velocity greater than "u"
D. Stays at rest on the inclined plane and will not side down.

Answer

To solve the problem, we need to analyze the motion of the object on the inclined plane under the influence of forces acting on it. Here's a step-by-step solution:

### Step 1: Understand the Forces Acting on the Object
When the object slides down the inclined plane with uniform velocity, it means that the net force acting on the object is zero. The forces acting on the object are:
- The gravitational force component acting down the incline: \( F_{\text{gravity}} = mg \sin \theta \)
- The frictional force acting up the incline: \( F_{\text{friction}} = f \)

Since the object is moving with uniform velocity, we have:
\[ F_{\text{gravity}} = F_{\text{friction}} \]
Thus,
\[ mg \sin \theta = f \]

### Step 2: Analyze the Situation When Pushed Up the Incline
Now, when the same object is pushed up the incline with an initial velocity \( u \), it will move against the gravitational force and friction. As it moves up, the forces acting on it will be:
- The gravitational force component acting down the incline: \( mg \sin \theta \)
- The frictional force acting down the incline (since it opposes the motion): \( f \)

### Step 3: Determine the Net Force Acting on the Object
The net force acting on the object when it is moving up the incline can be expressed as:
\[ F_{\text{net}} = -mg \sin \theta - f \]
Since we already established that \( f = mg \sin \theta \), we can substitute this into the equation:
\[ F_{\text{net}} = -mg \sin \theta - mg \sin \theta = -2mg \sin \theta \]

### Step 4: Analyze the Motion of the Object
As the object moves up the incline, it will decelerate due to the net force acting against its motion. Eventually, it will come to a stop when its velocity becomes zero. At this point, all the initial kinetic energy will have been converted into work done against the gravitational and frictional forces.

### Step 5: Determine What Happens Next
Once the object comes to a stop, the forces acting on it will again be balanced since the frictional force will equal the gravitational force component acting down the incline. Therefore, the object will not slide down the incline but will remain at rest.

### Conclusion
The correct option is that the object stays at rest on the inclined plane and does not slide down.