JEE PYQ: Motion in a Straight Line - Question ID c4abc9f8cbe6 (JEE Main 2007)

In kinematics, the displacement $x(t)$ of a particle moving along a straight line is obtained by integrating its velocity $v(t)$ with respect to time. The fundamental relation is: $x(t) = \int_{0}^{t} v(t') \, dt'.$ Given that the particle starts at $x = 0$ when $t = 0$, the constant of integration is zero. The velocity function provided is: $v(t) = v_0 + g t + f t^2.$ Our goal is to compute the displacement at $t = 1$ by integrating this velocity expression.

1. Write the velocity function: $v(t) = v_0 + g t + f t^2.$ 2. Integrate $v(t)$ from $0$ to $t$ to find the displacement $x(t)$: $x(t) = \int_{0}^{t} v(t') \, dt' = \int_{0}^{t} \left(v_0 + g t' + f t'^2\right) dt'.$ 3. Break the integral into three parts: $x(t) = \int_{0}^{t} v_0 \, dt' + \int_{0}^{t} g t' \, dt' + \int_{0}^{t} f t'^2 \, dt'.$ 4. Compute each integral separately: - $\int_{0}^{t} v_0 \, dt' = v_0 t$, - $\int_{0}^{t} g t' \, dt' = g \int_{0}^{t} t' \, dt' = g \left[\frac{t'^2}{2}\right]_0^t = \frac{g t^2}{2}$, - $\int_{0}^{t} f t'^2 \, dt' = f \int_{0}^{t} t'^2 \, dt' = f \left[\frac{t'^3}{3}\right]_0^t = \frac{f t^3}{3}$. 5. Combine the results: $x(t) = v_0 t + \frac{g t^2}{2} + \frac{f t^3}{3}.$ 6. Evaluate the displacement at $t = 1$: $x(1) = v_0 (1) + \frac{g (1)^2}{2} + \frac{f (1)^3}{3} = v_0 + \frac{g}{2} + \frac{f}{3}.$ 7. Compare with the given options. The expression $v_0 + \frac{g}{2} + \frac{f}{3}$ matches option C.

- Misidentifying the integral of $t^2$: Students often confuse $\int t^2 \, dt = \frac{t^3}{3}$ with $\frac{t^2}{2}$ or $t^3$. This leads to incorrect coefficients for $f$. - Forgetting the constant of integration: Since $x = 0$ at $t = 0$, the constant is zero, but overlooking this can cause confusion. - Evaluating at $t = 1$ incorrectly: Some students substitute $t = 1$ too early, before integrating, which disrupts the calculation. - Exam Tip: Always integrate the velocity function fully before substituting numerical values for time. Double-check the integration of polynomial terms to avoid arithmetic errors.