JEE PYQ: Motion in a Straight Line - Question ID 2ee3299549ea (JEE Main 2026)

This problem involves motion in a straight line with two distinct phases:

1. Free-fall phase (0–2 s): The paratrooper falls under gravity alone. We use the kinematic relation for uniformly accelerated motion: $s = ut + \tfrac{1}{2} a t^2, \quad v = u + a t.$ Here $u=0$, $a=g=10\ \mathrm{m/s^2}$.
2. Decelerated descent (after 2 s): The parachute opens and the paratrooper decelerates at $3\ \mathrm{m/s^2}$ until reaching $5\ \mathrm{m/s}$ at $10\ \mathrm{m}$ above the ground. We again use the kinematic relation $v^2 = v_0^2 + 2a s$ with $a=-3\ \mathrm{m/s^2}$ (deceleration).

The total height of the airplane is the sum of the distance fallen in the free-fall phase, the distance fallen while decelerating, and the final $10\ \mathrm{m}$ above ground.

1. Free-fall phase (0–2 s):
   Initial velocity $u=0$, acceleration $g=10\ \mathrm{m/s^2}$, time $t=2\ \mathrm{s}$.
   • Distance fallen: $s_1 = \tfrac{1}{2} g t^2 = \tfrac{1}{2}\times10\times2^2 = 20\ \mathrm{m}.$
   • Velocity at the end of free fall: $v_1 = g t = 10\times2 = 20\ \mathrm{m/s}.$
2. Decelerated descent phase:
   The paratrooper now decelerates at $3\ \mathrm{m/s^2}$ until reaching $5\ \mathrm{m/s}$ at $10\ \mathrm{m}$ above ground. Let $s_2$ be the distance fallen during this phase.
   • Use the kinematic relation $v^2 = v_1^2 + 2a s_2$ with $v=5\ \mathrm{m/s}$, $v_1=20\ \mathrm{m/s}$, $a=-3\ \mathrm{m/s^2}$.
   • Rearrange: $5^2 = 20^2 + 2(-3)s_2$ $25 = 400 - 6s_2$ $6s_2 = 400 - 25 = 375$ $s_2 = \frac{375}{6} = 62.5\ \mathrm{m}.$
3. Total height of the airplane:
   $H = s_1 + s_2 + 10 = 20 + 62.5 + 10 = 92.5\ \mathrm{m}.$

1. Sign of acceleration: Many students forget that deceleration means $a=-3\ \mathrm{m/s^2}$ in the upward-positive convention. 2. Final height: The $10\ \mathrm{m}$ above ground must be added to the distances fallen, not subtracted. 3. Velocity at parachute opening: One must compute $v_1=20\ \mathrm{m/s}$ before applying the deceleration relation.

Double-check each phase separately and sum the distances carefully to avoid off-by-ten errors.