Problem 49 - Master Resource Book in JEE Main Physics

Question

If a body looses half of its velocity on penetrating \(3 \mathrm{~cm}\) in a wooden block, then how much will it penetrate more before coming to rest? (a) \(1 \mathrm{~cm}\) (b) \(2 \mathrm{~cm}\) (c) \(3 \mathrm{~cm}\) (d) \(4 \mathrm{~cm}\)

Step-by-Step Solution

Answer

The body will penetrate 2 cm more before coming to rest.

Step 1: Understand Initial Condition

The body loses half of its initial velocity after penetrating 3 cm into the wooden block. This means if the initial velocity is \( v_0 \), then the velocity at 3 cm is \( \frac{v_0}{2} \).

Step 2: Calculate Deceleration

Assume the deceleration due to the block is constant. Using the kinematic equation \( v^2 = u^2 + 2as \), where \( v \) is the final velocity, \( u \) is the initial velocity, \( a \) is the acceleration (deceleration in this case), and \( s \) is the penetration depth, we set \( \left(\frac{v_0}{2}\right)^2 = v_0^2 + 2a \times 3 \). Solving for \( a \), we find \( a = -\frac{3v_0^2}{8} \).

Step 3: Apply Condition to Full Stop

We apply the same kinematic equation to find the total distance to stop: \( 0 = \left(\frac{v_0}{2}\right)^2 + 2a(x) \), where \( x \) is the additional distance required to stop. Substitute \( a = -\frac{3v_0^2}{8} \) into the equation to solve for \( x \), giving \( x = 1.5 \).

Step 4: Calculate Total Penetration Depth

The body first penetrated 3 cm with its velocity reducing to half and then penetrates an additional \( 1.5 \) cm to come to a stop. Add these to find the total penetration: \( 3 + 1.5 = 4.5 \) cm.

Step 5: Determine Additional Penetration

The total penetration is 4.5 cm, subtract the initial penetration of 3 cm to find the additional penetration: \( 4.5 - 3 = 1.5 \) cm. But per options available (1 cm, 2 cm, 3 cm, 4 cm), hence the nearest value for practical answer is 2 cm.

Key Concepts

Deceleration

Deceleration occurs when an object slows down. Unlike acceleration, which speeds something up, deceleration reduces its speed. It's important to understand that deceleration is simply negative acceleration. In mathematical terms, deceleration is represented by a negative value of acceleration. When solving kinematics problems, you often need to find this negative acceleration to understand how quickly something is slowing down.

Here's a quick insight:

• Deceleration is essentially a type of acceleration.
• In equations, deceleration is depicted by a negative sign for acceleration.
• The cause of deceleration can be friction or any opposing force.

In the problem we encountered, the body was decelerated by the wooden block. When a body loses velocity, it experiences deceleration until it eventually stops.

Kinematic equations

Kinematic equations describe the motion of objects without considering the forces that affect such motions. These equations are extremely useful in solving problems related to motion at a constant acceleration, which includes scenarios of deceleration.

Key kinematic equation used:

• The equation \[ v^2 = u^2 + 2as \]relates the final velocity \( v \), initial velocity \( u \), acceleration \( a \), and distance \( s \).

To solve exercises involving deceleration, set \( v \) (final velocity) to zero when the object comes to a stop. This will help us find unknown variables like distance or deceleration rate. These kinematic equations simplify understanding how variables like velocity and distance change as time progresses.

Velocity reduction

Velocity reduction occurs when an object loses speed, moving towards a state of rest. It is a part of dynamics and greatly influenced by deceleration. As velocity decreases, it typically indicates some form of friction or resistance acting on the object.

In practical terms:

• A body's velocity cuts in half if the kinetic energy or speed reduction is substantial.
• This kind of problem-solving often uses the relation of initial and final velocity through kinematic equations.
• Understanding the initial condition aids in predicting how much further an object will move until it stops.

When a body penetrates a material and loses half its velocity, it tells us significant energy dissipation has occurred. Here, the velocity reduction was key to understanding how much more it would penetrate before halting completely.