Physics Q45 – JEE-MAIN 2026

The equation of motion of a particle is given by

x = a \sin (50 t + \frac{π}{3})

cm. The particle will come to rest at time

t_{1}

, and it will have zero acceleration at time

t_{2}

. The

t_{1}

and

t_{2}

, respectively are _______.

Clue 1 of 3: Understanding SHM

Recall the definitions of velocity and acceleration for a particle undergoing Simple Harmonic Motion (SHM) from its displacement equation.

🧠 Full Solution Path

Step 1: Calculate Velocity and Acceleration✦ Active

The given displacement is $x = a \sin (50 t + \frac{π}{3})$ . Differentiate $x$ with respect to time to find velocity $v$ , and then differentiate $v$ to find acceleration $a^{'}$ .

v = \frac{d x}{d t} = 50 a \cos (50 t + \frac{π}{3})

a^{'} = \frac{d v}{d t} = - 2500 a \sin (50 t + \frac{π}{3})

Step 2: Determine

t_{1}

(time at rest)○ Expand

The particle comes to rest when its velocity $v = 0$ . Set the velocity equation to zero and solve for the smallest positive time $t_{1}$ .

50 a \cos (50 t_{1} + \frac{π}{3}) = 0 ⟹ \cos (50 t_{1} + \frac{π}{3}) = 0

For the first positive time,

50 t_{1} + \frac{π}{3} = \frac{π}{2}

50 t_{1} = \frac{π}{2} - \frac{π}{3} = \frac{π}{6} ⟹ t_{1} = \frac{π}{300} s

💡 Teacher's Secret Hint

Remember that $\cos θ = 0$ when $θ = (2 n + 1) \frac{π}{2}$ . Choose $n = 0$ for the smallest positive time.

Step 3: Determine

t_{2}

(time at zero acceleration)○ Expand

The acceleration is zero when $a^{'} = 0$ . Set the acceleration equation to zero and solve for the smallest positive time $t_{2}$ .

- 2500 a \sin (50 t_{2} + \frac{π}{3}) = 0 ⟹ \sin (50 t_{2} + \frac{π}{3}) = 0

For the first positive time,

50 t_{2} + \frac{π}{3} = π

(since

50 t_{2} + \frac{π}{3} = 0

would yield a negative

t_{2}

50 t_{2} = π - \frac{π}{3} = \frac{2 π}{3} ⟹ t_{2} = \frac{2 π}{150} = \frac{π}{75} s

Thus,

t_{1} = \frac{π}{300} s

and

t_{2} = \frac{π}{75} s

💡 Teacher's Secret Hint

Remember that $\sin θ = 0$ when $θ = n π$ . Choose $n = 1$ to get the smallest positive time for $t_{2}$ in this case.

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Physics Question 45 – JEE-MAIN 2026

Clue 1 of 3: Understanding SHM

🧠 Full Solution Path