Problem 333 - JEE Physics

Question

Same force acts on two bodies of different masses \(2 \mathrm{~kg}\) and \(4 \mathrm{~kg}\) initially at rest. the ratio of times required to acquire same final velocity is (A) \(2: 1\) (B) \(1: 2\) (C) \(1: 1\) (D) \(4: 16\)

Step-by-Step Solution

Answer

The ratio of times required for the two bodies to reach the same final velocity is \(1:2\), which corresponds to option (B).

Step 1: Understanding the given problem

We are given that the same force acts on two bodies with masses 2 kg and 4 kg which are initially at rest. We need to find the ratio of time taken for these two bodies to achieve the same final velocity.

Step 2: Using force equation and kinematic equation

We will use the equation F = ma (Force = mass * acceleration) to find the acceleration of each body, where 'm' is the mass and 'a' is the acceleration. Additionally, we know that final velocity (v) can be determined using the kinematic equation: \(v = u + at\), where 'u' is the initial velocity, 'a' is the acceleration, and 't' is the time taken.

Step 3: Finding acceleration for each body using force equation

Since the same force acts on both bodies, we can denote the force as F. For the first body with mass 2 kg and acceleration \(a_1\): \(F = m_1a_1\) \(a_1 = \frac{F}{m_1}\) For the second body with mass 4 kg and acceleration \(a_2\): \(F = m_2a_2\) \(a_2 = \frac{F}{m_2}\)

Step 4: Using the kinematic equation for final velocity of each body

We are given that both bodies reach the same final velocity 'v'. Since both bodies are initially at rest, their initial velocity (u) is 0. For the first body with acceleration \(a_1\) and time \(t_1\): \(v = 0 + a_1t_1\) \(t_1 = \frac{v}{a_1}\) For the second body with acceleration \(a_2\) and time \(t_2\): \(v = 0 + a_2t_2\) \(t_2 = \frac{v}{a_2}\)

Step 5: Finding the ratio of times required to reach the same final velocity

Now, we need to find the ratio \(T = \frac{t_1}{t_2}\) Substituting the values of \(t_1\) and \(t_2\) using the information obtained in step 4: \(T = \frac{\frac{v}{a_1}}{\frac{v}{a_2}}\) \(T = \frac{a_2}{a_1}\) Substituting the values of \(a_1\) and \(a_2\) determined in step 3: \(T = \frac{\frac{F}{m_2}}{\frac{F}{m_1}}\) \(T = \frac{m_1}{m_2}\) Finally, substituting the values of \(m_1 = 2\ \mathrm{kg}\) and \(m_2 = 4\ \mathrm{kg}\): \(T = \frac{2\ \mathrm{kg}}{4\ \mathrm{kg}} = \frac{1}{2}\) So, the ratio of times required to reach the same final velocity is \(\boxed{1:2}\) which corresponds to option (B).

Key Concepts

Force and Mass Relationship

The force and mass relationship is a fundamental part of Newton's Second Law of Motion, which can be expressed as the equation \( F = ma \). This formula tells us that the force applied to an object is equal to the mass of the object multiplied by its acceleration. In simpler terms, if you apply the same amount of force to two objects of different masses, the one with less mass will accelerate more. This concept is crucial in understanding how different masses respond to the same force.
The exercise we're exploring helps illustrate this relationship. We have two bodies, one with a mass of 2 kg and the other with 4 kg. Both are subjected to the same force. According to \( F = ma \), the acceleration experienced by each body can be found by rearranging the formula to \( a = \frac{F}{m} \). Therefore, for a constant force, a heavier object (greater mass \( m \)) will have less acceleration, while a lighter object will experience more. This core understanding of how force and mass interact is central to solving the problem.

Kinematics Equations

Kinematics equations describe the motion of objects based on their initial velocity, acceleration, and the time involved. One common kinematic equation is \( v = u + at \), where:

• \(v\) is the final velocity,
• \(u\) is the initial velocity,
• \(a\) is the acceleration,
• \(t\) is the time taken.

These equations are particularly useful in problems involving uniformly accelerated motion, like the one in the exercise.
In the given scenario, both bodies start from rest, which means the initial velocity \( u = 0 \). The equation simplifies to \( v = at \). This equation allows us to find the time taken to reach a certain final velocity, provided we know the acceleration, which in turn, depends on the force and mass relationship as explained earlier. By applying these fundamental kinematics concepts, we can determine how long each body takes to reach the desired speed.

Acceleration Calculation

Acceleration is the rate at which an object changes its velocity. It is a key concept when solving problems involving forces and motion. In the context of our exercise, it's essential to calculate the acceleration for each body to find the time it takes to acquire the same final velocity.
Given the force \( F \) is constant for both bodies, we use the rearranged version of Newton's Second Law, \( a = \frac{F}{m} \), to determine the acceleration:

• For the 2 kg body, the acceleration \( a_1 = \frac{F}{2} \).
• For the 4 kg body, the acceleration \( a_2 = \frac{F}{4} \).

This calculation step highlights the force-to-mass dependency, showing that the lighter body (2 kg) will accelerate more than the heavier one (4 kg) given the same force. Understanding how to calculate acceleration in different scenarios is vital for analyzing motion in physics.

Time Ratio in Motion

The time ratio in motion is a way to compare how long it takes different objects to reach a particular state under certain conditions. In our problem, we want to find out how the times compare for two masses achieving the same final velocity when acted upon by the same force.
First, identify the expression for time in terms of acceleration and velocity from the kinematic equation \( v = at \):
\( t = \frac{v}{a} \). By substituting the expressions for acceleration (\( a_1\) and \( a_2\)) calculated previously:

• The time for the 2 kg body, \( t_1 = \frac{v}{a_1} = \frac{v}{F/2} \).
• The time for the 4 kg body, \( t_2 = \frac{v}{a_2} = \frac{v}{F/4} \).

Thus, their time ratio is given by \( \frac{t_1}{t_2} = \frac{a_2}{a_1} \) which simplifies to \( \frac{m_1}{m_2} = \frac{2}{4} = \frac{1}{2} \). Hence, it requires half as much time for the lighter body to reach the same velocity, emphasizing how mass affects acceleration and required time under identical forces.