JEE PYQ: Motion in a Straight Line - Question ID b5b1c738e48a (JEE Main 2021)

Core Formula & Concept:

When a particle moves with constant acceleration along a straight line, its motion is governed by three fundamental kinematic relations:

1. Position as a function of time: $x(t) = x_0 + v_0 t + \tfrac{1}{2} a t^2$ This is a quadratic function of time, so its graph is a parabola. 2. Velocity as a function of time: $v(t) = v_0 + a t$ This is a linear function of time, so its graph is a straight line with slope equal to the constant acceleration \(a\). 3. Acceleration as a function of time: $a(t) = a$ Since the acceleration is constant, its graph is a horizontal straight line.

The question asks which set of three graphs (position, velocity, acceleration) correctly represents these shapes for motion with constant acceleration.

Step-by-Step Derivation:

Step 1: Identify the expected shapes
- Position vs time: parabola (quadratic in \(t\)). - Velocity vs time: straight line (linear in \(t\)). - Acceleration vs time: horizontal line (constant).

Step 2: Examine each option

Option A
- Position graph: straight line (linear) ⇒ incorrect (should be parabolic). - Velocity graph: straight line ⇒ correct. - Acceleration graph: horizontal line ⇒ correct. Since the position graph is wrong, Option A is invalid.

Option B
- Position graph: parabola (quadratic) ⇒ correct. - Velocity graph: straight line ⇒ correct. - Acceleration graph: horizontal line ⇒ correct. All three graphs match the expected shapes. Option B is valid.

Option C
- Position graph: parabola ⇒ correct. - Velocity graph: parabola ⇒ incorrect (should be linear). - Acceleration graph: horizontal line ⇒ correct. The velocity graph is wrong, so Option C is invalid.

Option D
- Position graph: parabola ⇒ correct. - Velocity graph: straight line ⇒ correct. - Acceleration graph: straight line with positive slope ⇒ incorrect (acceleration must be constant, hence horizontal). The acceleration graph is wrong, so Option D is invalid.

Conclusion
Only Option B satisfies all three kinematic graphs for motion with constant acceleration.

Common Traps & Exam Tip:

1. Confusing position and velocity graphs: Many students mistake the parabolic position graph for the velocity graph. Remember: velocity is linear, position is quadratic. 2. Acceleration graph slope: A sloped acceleration graph implies changing acceleration, which contradicts the “constant acceleration” condition. 3. Graph concavity: The parabola for position must open upwards if acceleration is positive, and downwards if negative. Ensure the concavity matches the sign of \(a\).

Exam Tip: Always sketch the three graphs mentally before looking at the options. This quick check can save precious time.