Problem 10 - Master Resource Book in JEE Main Physics

Question

42\. A particle moves along \(x\)-axis as $$ x=4(t-2)+a(t-2)^{2} $$ Which of the following is true? (a) The initial velocity of particle is 4 (b) The acceleration of particle is \(2 \mathrm{a}\) (c) The particle is at origin at \(t=0\) (d) None of the above

Step-by-Step Solution

Answer

(b) The acceleration of particle is \(2a\).

Step 1: Substitute Initial Time in Position Equation

To find the initial position of the particle at time \( t = 0 \), substitute \( t = 0 \) in the given equation: \[ x = 4(0-2) + a(0-2)^2 \] Simplifying this gives: \[ x = -8 + 4a \] The particle is not at the origin if this expression is not zero, ruling out option (c).

Step 2: Find Initial Velocity by Differentiating Position

Differentiate the position equation to find the velocity function. The position function is: \[ x = 4(t-2) + a(t-2)^2 \] Differentiate with respect to \( t \) to find velocity \( v \): \[ v = \frac{d}{dt}[4(t-2)] + \frac{d}{dt}[a(t-2)^2] \] \[ v = 4 + 2a(t-2) \] To find the initial velocity, substitute \( t = 0 \): \[ v = 4 + 2a(0-2) = 4 - 4a \] The initial velocity is not 4, ruling out option (a).

Step 3: Find Acceleration by Differentiating Velocity

Differentiate the velocity equation to find the acceleration function. The velocity function is: \[ v = 4 + 2a(t-2) \] Differentiate with respect to \( t \) to find acceleration \( a \): \[ a = \frac{d}{dt}[4 + 2a(t-2)] = 2a \] This matches option (b): the acceleration of the particle is indeed \( 2a \).

Step 4: Conclude based on Analysis

Based on the calculations from the previous steps, option (b) is correct: the acceleration of the particle is \( 2a \). Options (a) and (c) are incorrect based on our derivations.

Key Concepts

Equations of Motion

In physics, the equations of motion describe the behavior of a moving object. They allow us to understand different characteristics such as position, velocity, and acceleration over time.

For this particle, the position along the x-axis is given by the equation: \[ x = 4(t-2) + a(t-2)^2 \] This equation can help us find where the particle is at any time \( t \).

Each term in the equation serves a specific purpose. The term \(4(t-2)\) shows linear motion while \(a(t-2)^2\) reveals how acceleration affects its movement.

By substituting different values of \(t\), we can analyze the particle's journey over time. This foundational equation is key for solving problems related to motion.

Differentiation in Physics

Differentiation is a process that helps us find rates of change, particularly useful in physics for finding velocity and acceleration from position.

To find the velocity, we differentiate the given position function with respect to time \( t \). This tells us how fast the position is changing. For our particle, the position function \( x = 4(t-2) + a(t-2)^2 \) becomes the velocity function:

• Velocity, \( v = \frac{d}{dt}[4(t-2)] + \frac{d}{dt}[a(t-2)^2] = 4 + 2a(t-2) \)

Differentiation helps us zoom into the motion, providing insights into the speed and direction changes over time.

Then, by differentiating the velocity function, \( v = 4 + 2a(t-2) \), we find the acceleration function which details the rate of change of velocity. Differentiation as a tool in physics paves the way for deep understanding of motion dynamics.

Initial Velocity

Initial velocity is the speed of the particle at the very start, when \( t = 0 \). It is a crucial concept because it sets the stage for future motion.

To find the initial velocity of our particle, we substitute \( t = 0 \) into the velocity function:

• Velocity, \( v = 4 + 2a(0-2) = 4 - 4a \)

Contrary to option (a), the initial velocity isn't 4 unless \( a = 0 \). This illustrates how any present initial acceleration impacts the starting velocity.

Understanding initial conditions helps predict the entire trajectory of a moving object and is thus a key part of solving any motion-related physics problem.

Acceleration Analysis

Acceleration analysis reveals how the velocity changes with time, informing us about the speed increments or decrements.

In the given problem, after differentiating the velocity function, \( v = 4 + 2a(t-2) \), we find that the acceleration is:

• Acceleration, \( a = \frac{d}{dt}[4 + 2a(t-2)] = 2a \)

This result aligns with option (b), confirming its correctness.

Acceleration indicates how forces might be acting on a particle, impacting motion over time. Analyzing acceleration allows one to understand and predict how an object's motion evolves and is fundamental in physics to explore dynamic behavior.