Question
A particle moves along a straight line such that its position is given by metres, where is in seconds. Find the time at which the particle momentarily comes to rest, and the displacement during the first 3 seconds. (Adapted from JEE Main 2023, Shift 2.)
Solution — Step by Step
Velocity is the rate of change of position. Differentiating once:
We want the instants where , since “momentarily at rest” means zero velocity.
Setting and dividing through by 6:
So the particle is at rest at and . Both are valid because the particle reverses direction twice on this path.
We need positions at to track the back-and-forth motion:
- m
- m
- m
- m
Displacement is final minus initial position, regardless of how many turns the particle takes:
Final answers: at rest at s and s; displacement in first 3 s = 9 m.
Why This Works
The trick examiners love here is the difference between displacement and distance travelled. Displacement only cares about the start and end points, so we can ignore the wiggling in between.
If they had asked for total distance, we would add m. Read the question word by word — JEE setters routinely swap these terms to catch tired students.
Alternative Method
We could use definite integration of speed:
But for displacement alone, the fundamental theorem of calculus gives directly. Same answer, less work.
Common Mistake
Students often report the answer as 11 m because they forget that the question asks for displacement, not distance. Always underline “displacement” or “distance” in the question before solving — this single habit saves 1 to 2 marks per kinematics PYQ.