Question

A block B, placed on a horizontal surface is pulled with initial velocity \(\mathrm{V}\). If the coefficient of kinetic friction between surface and block is \(\mu\), than after how much time, block will come to rest? (A) (v/g) (B) \((\mathrm{g} / \mathrm{v})\) (C) \((\mathrm{g} / \mathrm{v})\) (D) \((\mathrm{v} / \mathrm{g})\)

Short answer

The time it takes for the block to come to rest is \(t = \frac{V}{\mu g}\).

Step-by-step solution

1. Understand the problem

The block B is initially moving with velocity V on a horizontal surface. The coefficient of kinetic friction between the block and the surface is μ. The frictional force will act on the block opposite to its motion, causing the block to come to rest eventually. We need to find the time it takes for the block to come to rest.

2. Write expressions for frictional force and acceleration

As the block is on a horizontal surface, the normal force acting on the block is equal to its weight mg (where m is the mass of the block and g is the acceleration due to gravity). The frictional force acting on the block is given by: \(F_f = \mu F_n\) where \(F_f\) is the frictional force, \(\mu\) is the coefficient of kinetic friction, and \(F_n\) is the normal force. Therefore, the frictional force acting in this case is: \(F_f = \mu mg\) Since the frictional force is acting opposite to the direction of the motion of the block, it causes a deceleration (negative acceleration) in the block. According to Newton's second law, the acceleration of the block is given by the force acting on it, divided by its mass: \(a = -\frac{F_f}{m}\)

3. Calculate deceleration

Substitute the expression of the frictional force from the previous step into the equation for acceleration: \(a = -\frac{\mu mg}{m}\) Simplify the expression by canceling the mass term, m: \(a = -\mu g\) The negative sign indicates that the acceleration is acting opposite to the direction of the motion, i.e., deceleration.

4. Find the stopping time

Use the deceleration equation, which is the final velocity equals initial velocity plus acceleration times time: \[v_f = v_i + at\] Since the block is coming to rest, the final velocity is \(0\), and the initial velocity is \(V\). As the block is decelerating, its acceleration is \(-\mu g\). Substitute these values into the equation: \(0 = V - \mu g t\) Rearrange the equation for t: \(t = \frac{V}{\mu g}\) Now we have found the time it takes for the block to come to rest. Comparing it with the available options, the correct answer is: (A) \(\frac{V}{g}\)