JEE PYQ: Motion in a Plane - Question ID 8992be460bd6 (JEE Main 2013)

In projectile motion, the motion of an object can be resolved into two independent components:

• Horizontal motion ($\widehat{i}$ direction): Uniform motion with constant velocity since no acceleration acts horizontally (ignoring air resistance).
• Vertical motion ($\widehat{j}$ direction): Uniformly accelerated motion under gravity ($g = 10 \, \text{m/s}^2$ downward).

The key formulas used are:

• Horizontal displacement: $x = u_x t$, where $u_x$ is the horizontal component of initial velocity.
• Vertical displacement: $y = u_y t - \frac{1}{2} g t^2$, where $u_y$ is the vertical component of initial velocity.

The trajectory equation is obtained by eliminating the time parameter $t$ from the above equations, yielding a relationship between $y$ and $x$.

Step 1: Identify the initial velocity components

The initial velocity vector is given as: $\vec{v} = \left( \widehat{i} + 2\widehat{j} \right) \, \text{m/s}$ Thus,

• Horizontal component: $u_x = 1 \, \text{m/s}$
• Vertical component: $u_y = 2 \, \text{m/s}$

Step 2: Express horizontal displacement as a function of time

Since horizontal motion is uniform: $x = u_x t = 1 \cdot t \implies t = x$

Step 3: Express vertical displacement as a function of time

Using the equation for vertical motion under gravity: $y = u_y t - \frac{1}{2} g t^2 = 2t - \frac{1}{2} \cdot 10 \cdot t^2 = 2t - 5t^2$

Step 4: Eliminate time $t$ to find the trajectory equation

From Step 2, we have $t = x$. Substitute this into the expression for $y$: $y = 2x - 5x^2$

Step 5: Compare with the given options

The derived trajectory equation is: $y = 2x - 5x^2$ This matches Option B. --- Common Traps & Exam Tip:

Students often make the following mistakes:

• Incorrect sign for gravity: Some take $g$ as positive, leading to $y = 2x + 5x^2$, which is incorrect. Always ensure the acceleration due to gravity acts downward (negative in standard coordinate systems).
• Miscounting coefficients: Misreading the initial velocity components (e.g., taking $u_x = 2$ and $u_y = 1$) leads to $y = \frac{1}{2}x - 5x^2$, which is not among the options but wastes time.
• Algebraic errors in substitution: Forgetting to replace $t$ with $x$ or squaring $x$ incorrectly can lead to wrong options like A or C.

Exam Tip: Always double-check the substitution step where $t$ is replaced with $x$. Verify the final equation by plugging in a small value of $x$ (e.g., $x = 0.2$) to see if $y$ matches the expected value from the original equations.