Problem 42 - Master Resource Book in JEE Main Physics

Question

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

The acceleration of the particle is \(2a\), so (b) is correct.

Step 1: Understand the Motion Equation

The given equation of motion of the particle is \[x = 4(t-2) + a(t-2)^2\]which describes how the position \(x\) changes with time \(t\). Identify from the equation the dependency of position on time and the parameters involved.

Step 2: Evaluate Initial Velocity

The term for velocity in the expression involves the coefficient of the linear term in time after converting the given form to the standard form of motion equations. Here, the coefficient of \(t\) in expansion \[x = 4t - 8 + a(t^2 - 4t + 4)\]will give the velocity function:\[v = 4 + 2a(t-2)\]At \(t = 0\), it's \[v(0) = 4 - 4a\], which means the initial velocity isn't 4 unless \(a = 0\).

Step 3: Calculate the Acceleration

Acceleration is the second derivative of position with respect to time. Analyze the equation for acceleration:\[a = \frac{d^2x}{dt^2}\]Using the velocity equation from Step 2, differentiate again:\[a = \frac{d}{dt} v = 2a\]Thus, the acceleration of the particle is \(2a\). This suggests that option (b) is correct.

Step 4: Determine Initial Position

Substitute \(t = 0\) into the position equation to see if the particle is at the origin:\[x(0) = 4(-2) + a(-2)^2 = -8 + 4a\]For the particle to be in the origin (i.e., \(x = 0\)), we need:\[-8 + 4a = 0 \]which implies \(a = 2\). Hence, without knowing \(a\), the particle is not at the origin for all values of \(a\).

Key Concepts

Particle Motion

Particle motion can be understood by examining how an object's position changes over time. In this context, we observe a particle moving along the x-axis. This movement is represented by a specific mathematical equation: \(x = 4(t-2) + a(t-2)^2\). The equation illustrates the relationship between time \(t\) and the position \(x\), influenced by the parameter \(a\). The equation is critical in analyzing how the particle travels as time progresses.

Key aspects of particle motion include:

• Understanding the motion equation which links the particle's position to time and other parameters.
• Analyzing how changes in these parameters affect the particle's movement.

By decomposing the motion equation, you gain insights into the velocity and acceleration of the particle, which are crucial for predicting future movement.

Initial Velocity

Initial velocity is essentially the speed of the particle at the beginning of the observation, or when \(t = 0\). To determine this from the given equation \(x = 4(t-2) + a(t-2)^2\), we first convert it to a more familiar form by expanding it. The initial velocity is revealed in the coefficient of the first-order term in time \(t\) within the expanded form:

The velocity as a function is found to be \(v = 4 + 2a(t-2)\), where the term "4" represents the constant speed aspect, and the term \(2a(t-2)\) suggests variations due to acceleration. When \(t = 0\), initial velocity simplifies to \(v(0) = 4 - 4a\).

• It shows that the initial velocity changes based on \(a\).
• For the initial speed to be exactly 4 units, \(a\) must be zero, which very specifically impacts the equation.

Acceleration Calculation

Acceleration is the rate of change of velocity over time. It provides us with insight on how quickly the particle is speeding up or slowing down. In mathematical terms, it is the second derivative of the position \(x\) with respect to time \(t\).

Given the equation for velocity as \(v = 4 + 2a(t-2)\), the acceleration \(a\) can be calculated by differentiating the velocity function with respect to time \(t\). This gives us a constant acceleration \(a = 2a\), showing that acceleration is directly proportional to \(a\) itself.

• It is significant because it describes how the force acting on the particle impacts its motion resulted through acceleration.
• The acceleration being \(2a\) explains why option b from the original choices is correct.

Position Equation

The position equation is a mathematical description of where the particle is at any given time. By inputting time \(t\), you can calculate the exact location of the particle along the x-axis. The initial set equation is \(x = 4(t-2) + a(t-2)^2\).

This equation is expanded to further demonstrate dependencies and results in \(x = 4t - 8 + a(t^2 - 4t + 4)\). When \(t = 0\), we directly substitute to find the initial position:

The calculation results in \(x(0) = -8 + 4a\). To verify if the particle is at origin at \(t = 0\), this needs to equal zero. Thus, we find \(4a = 8\) or \(a = 2\) for the particle to sit at the origin initially. Without precise data for \(a\), you cannot generally claim that the particle is at the origin.