JEE PYQ: Motion in a Plane - Question ID b2fbec8ee5e3 (JEE Main 2025)

Core Formula & Concept:

In projectile motion, the velocity of the ball can be resolved into two perpendicular components:

• Horizontal component: $v_x = v \cos \theta$ (remains constant throughout the motion, since no acceleration acts horizontally).
• Vertical component: $v_y = v \sin \theta$ (changes due to gravity, becoming zero at the highest point).

The kinetic energy ($KE$) of the ball is given by: $KE = \frac{1}{2} m v^2$ where $m$ is the mass and $v$ is the speed of the ball.

At the highest point of the trajectory, the vertical component of velocity becomes zero, so the speed of the ball is only due to its horizontal component. The decrease in kinetic energy is the difference between the initial kinetic energy and the kinetic energy at the highest point.

Step-by-Step Derivation:

Step 1: Convert mass to SI units
Given mass $m = 100 \, \text{g} = 0.1 \, \text{kg}$. Step 2: Resolve initial velocity into components
Initial velocity $v = 20 \, \text{m/s}$ at $\theta = 60^\circ$.

• Horizontal component: $v_x = v \cos 60^\circ = 20 \times \frac{1}{2} = 10 \, \text{m/s}$.
• Vertical component: $v_y = v \sin 60^\circ = 20 \times \frac{\sqrt{3}}{2} = 10\sqrt{3} \, \text{m/s}$.

Step 3: Compute initial kinetic energy
Initial speed $v = 20 \, \text{m/s}$, so: $KE_{\text{initial}} = \frac{1}{2} m v^2 = \frac{1}{2} \times 0.1 \times (20)^2 = \frac{1}{2} \times 0.1 \times 400 = 20 \, \text{J}$ Step 4: Compute kinetic energy at the highest point
At the highest point, $v_y = 0$, so the speed is $v_x = 10 \, \text{m/s}$. $KE_{\text{highest}} = \frac{1}{2} m v_x^2 = \frac{1}{2} \times 0.1 \times (10)^2 = \frac{1}{2} \times 0.1 \times 100 = 5 \, \text{J}$ Step 5: Calculate the decrease in kinetic energy
$\Delta KE = KE_{\text{initial}} - KE_{\text{highest}} = 20 \, \text{J} - 5 \, \text{J} = 15 \, \text{J}$

Common Traps & Exam Tip:

• Incorrect mass unit: Students often forget to convert mass from grams to kilograms, leading to wrong kinetic energy values. Always ensure SI units are used.
• Misidentifying velocity at highest point: Some assume the speed at the highest point is zero, ignoring the horizontal component. Remember, only the vertical component becomes zero.
• Energy conservation confusion: While total mechanical energy (kinetic + potential) is conserved, kinetic energy alone decreases as potential energy increases. Do not confuse the two.

Final Answer: The decrease in kinetic energy is 15 J, which corresponds to option C.