JEE PYQ: Motion in a Straight Line - Question ID 1f68aae45b62 (JEE Main 2021)

This problem involves the motion of two spherical balls under constant gravitational acceleration ($g = 10 \, \text{m/s}^2$ downward). The key concepts and formulas used are:

• Kinematic Equation for Vertical Motion: For an object thrown upward with initial velocity $u$, its displacement $s$ at time $t$ is given by: $s = ut - \frac{1}{2}gt^2$ This equation accounts for the deceleration due to gravity.
• Relative Motion: Since both balls are thrown along the same vertical line with the same initial velocity but at different times, we analyze their positions as functions of time and find the instant when they meet (collide).
• Time Interval Consideration: The second ball is thrown $3 \, \text{s}$ after the first. Thus, if $t$ is the time elapsed since the first ball was thrown, the second ball has been in motion for $(t - 3) \, \text{s}$.

Let’s define:

• $u = 35 \, \text{m/s}$ (initial velocity of both balls)
• $g = 10 \, \text{m/s}^2$ (acceleration due to gravity)
• $t$ = time elapsed since the first ball was thrown (in seconds)
• $h_1(t)$ = height of the first ball at time $t$
• $h_2(t)$ = height of the second ball at time $t$

The second ball is thrown at $t = 3 \, \text{s}$, so for $t < 3$, $h_2(t) = 0$ (not yet thrown). For $t \geq 3$, the second ball has been in motion for $(t - 3) \, \text{s}$.

Using the kinematic equation for vertical motion:

The balls collide when $h_1(t) = h_2(t)$. Thus, we set:

Simplify the right-hand side (RHS):

Substitute back into the equation:

Combine like terms on the RHS:

Subtract $(35t - 5t^2)$ from both sides:

Thus, the balls collide at $t = 5 \, \text{s}$ after the first ball is thrown.

Now, compute the height at $t = 5 \, \text{s}$ using $h_1(t)$:

Verification: Check $h_2(t)$ at $t = 5 \, \text{s}$:

Both heights match, confirming the collision at $50 \, \text{m}$.

1. Incorrect Time Reference: Students often confuse the time variables. The second ball’s motion must be referenced to its own launch time ($t - 3$), not the absolute time $t$. Failing to adjust for the $3 \, \text{s}$ delay leads to incorrect equations.
2. Sign Errors in Kinematic Equation: The equation $s = ut - \frac{1}{2}gt^2$ assumes upward as positive. Some students mistakenly use $+ \frac{1}{2}gt^2$, which reverses the direction of gravity and yields wrong results.
3. Algebraic Mistakes: Expanding $(t - 3)^2$ incorrectly or miscombining terms during simplification can lead to wrong values for $t$. Double-check each algebraic step.
4. Ignoring Physical Constraints: The collision must occur after the second ball is thrown ($t \geq 3$). If solving yields $t < 3$, it implies no collision, which is not the case here. Always verify the solution’s physical validity.

Exam Tip: For problems involving relative motion with time delays, always define a clear time reference (e.g., $t = 0$ when the first object is launched). Write separate equations for each object’s position as a function of $t$, then equate them to find the collision time. This systematic approach minimizes errors.