The angle between frictional force and the instantaneous velocity of the body moving over a rough surface is

The correct answer is π (180°).

To understand why, we need to carefully analyze the nature of kinetic (sliding) friction and how it relates to the motion of a body.

Step 1: Understand What Friction Is

When a body moves over a rough surface, a resistive force acts on it due to the contact between the two surfaces. This resistive force is called kinetic friction (or sliding friction). Its fundamental property is that it always opposes the relative motion of the body.

Step 2: Identify the Direction of Instantaneous Velocity

At any given instant, the body is moving in a specific direction. The instantaneous velocity is a vector pointing in the direction of the body's motion at that instant. Let's say the body is moving to the right, so the velocity vector v points to the right (→).

Step 3: Identify the Direction of Friction

Since kinetic friction always acts to oppose the motion, it acts in the direction exactly opposite to the instantaneous velocity. If the body moves to the right (→), friction acts to the left (←).

Step 4: Find the Angle Between the Two Vectors

Now we have two vectors:

• Instantaneous velocity v → (pointing in the direction of motion)

• Frictional force f ← (pointing exactly opposite to the direction of motion)

Two vectors that point in exactly opposite directions form an angle of 180° between them, which in radians is:

180°=\pi \text{radians}

Step 5: Mathematical Confirmation

Recall the dot product formula for two vectors A and B:

\mathbf{A}·\mathbf{B}=\left|\mathbf{A}\right|\left|\mathbf{B}\right|cos\theta

Since friction force f is directly opposite to velocity v, we can write:

\mathbf{f}·\mathbf{v}=-\left|\mathbf{f}\right|\left|\mathbf{v}\right|

Substituting into the dot product formula:

-\left|\mathbf{f}\right|\left|\mathbf{v}\right|=\left|\mathbf{f}\right|\left|\mathbf{v}\right|cos\theta

cos\theta =-1

\theta ={cos}^{-1}(-1)=180°=\pi

Conclusion: The angle between the frictional force and the instantaneous velocity of a body moving over a rough surface is always π radians (180°), because friction always acts in the direction directly opposite to the motion of the body. This is why friction decelerates the object — a force at 180° to the velocity removes kinetic energy from the system.