Physics Q84 – AP-EAMCET 2025

Home Physics Motion in a Straight LineQuestion #316

For a particle moving in x-y plane, if at any instant of time '

t

', (in second) its displacements (in metre) are

x = 2 t^{2} - t

and

y = 4 t^{2} - 4 t

, then the velocity of the particle at a time

t = 1 s

Clue 1 of 3: Kinematics of Particle Motion

Velocity is the rate of change of displacement with respect to time. For motion in a plane, the velocity can be found by differentiating the position vector components.

🧠 Full Solution Path

Step 1: Understand the given displacement equations✦ Active

The position of the particle in the x-y plane is given by its displacement components as functions of time:

x (t) = 2 t^{2} - t

y (t) = 4 t^{2} - 4 t

We need to find the magnitude of the velocity at $t = 1 s$ .

Step 2: Determine the velocity components by differentiation○ Expand

Velocity is the time derivative of displacement. We differentiate $x (t)$ and $y (t)$ with respect to time $t$ to find the velocity components $v_{x} (t)$ and $v_{y} (t)$ .

v_{x} = \frac{d x}{d t} = \frac{d}{d t} (2 t^{2} - t) = 4 t - 1

v_{y} = \frac{d y}{d t} = \frac{d}{d t} (4 t^{2} - 4 t) = 8 t - 4

💡 Teacher's Secret Hint

Remember the power rule for differentiation: $\frac{d}{d t} (a t^{n}) = n a t^{n - 1}$ and $\frac{d}{d t} (c t) = c$ .

Step 3: Calculate the velocity components at the specified time○ Expand

Substitute $t = 1 s$ into the expressions for $v_{x} (t)$ and $v_{y} (t)$ :

v_{x} (t = 1 s) = 4 (1) - 1 = 3 m/s

v_{y} (t = 1 s) = 8 (1) - 4 = 4 m/s

💡 Teacher's Secret Hint

Ensure units are consistent throughout the calculation. Here, displacement is in meters and time in seconds, so velocity will be in m/s.

Step 4: Calculate the magnitude of the velocity○ Expand

The magnitude of the velocity vector $\vec{v} = v_{x} \hat{i} + v_{y} \hat{j}$ is given by $v = \sqrt{v_{x}^{2} + v_{y}^{2}}$ .

v = \sqrt{(3 m/s)^{2} + (4 m/s)^{2}}

v = \sqrt{9 + 16}

v = \sqrt{25}

v = 5 m/s

The velocity of the particle at $t = 1 s$ is $5 {m s}^{- 1}$ .

💡 Teacher's Secret Hint

This is a classic 3-4-5 Pythagorean triplet, which often appears in physics problems involving vector magnitudes.

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Physics Question 84 – AP-EAMCET 2025

Clue 1 of 3: Kinematics of Particle Motion

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