Physics Q28 – JEE-MAIN 2025

Home Physics Motion in a Straight LineQuestion #332

A particle moves along the

x

-axis and has its displacement

x

varying with time

t

according to the equation:

x = c_{0} (t^{2} - 2) + c (t - 2)^{2}

where

c_{0}

and

c

are constants of appropriate dimensions. Then, which of the following statements is correct?

Clue 1 of 3: Kinematic Definitions

Recall that velocity is the rate of change of displacement, and acceleration is the rate of change of velocity.

🥷

Ninja Strategy•Term-by-Term Differentiation

Recognize that the acceleration is the second derivative of displacement, and since the displacement equation is a sum of two terms, the acceleration will be the sum of the second derivatives of each term.

🧠 Full Solution Path

Step 1: Calculate the velocity of the particle.✦ Active

The displacement of the particle is given by $x = c_{0} (t^{2} - 2) + c (t - 2)^{2}$ . To find the velocity $v$ , we differentiate $x$ with respect to time $t$ :

v = \frac{d x}{d t} = \frac{d}{d t} [c_{0} (t^{2} - 2) + c (t - 2)^{2}]

v = c_{0} (2 t) + c \cdot 2 (t - 2) \cdot 1

v = 2 c_{0} t + 2 c (t - 2)

The initial velocity (at $t = 0$ ) is $v (0) = 2 c_{0} (0) + 2 c (0 - 2) = - 4 c$ . Therefore, option 1 is incorrect.

Step 2: Calculate the acceleration of the particle.○ Expand

To find the acceleration $a$ , we differentiate the velocity $v$ with respect to time $t$ :

a = \frac{d v}{d t} = \frac{d}{d t} [2 c_{0} t + 2 c (t - 2)]

a = 2 c_{0} (1) + 2 c (1)

a = 2 c_{0} + 2 c

a = 2 (c_{0} + c)

💡 Teacher's Secret Hint

Notice that the acceleration is constant, as expected for a displacement equation that is quadratic in time.

Step 3: Compare the result with the given options.○ Expand

The calculated acceleration is $a = 2 (c_{0} + c)$ . Comparing this with the given options: Option 2: "the acceleration of the particle is $2 c_{0}$ " (Incorrect) Option 3: "the acceleration of the particle is $2 c$ " (Incorrect) Option 4: "the acceleration of the particle is $2 (c + c_{0})$ " (Correct) Thus, the correct statement is option 4.

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Physics Question 28 – JEE-MAIN 2025

Clue 1 of 3: Kinematic Definitions

🧠 Full Solution Path