At an instant t , the coordinates of a particle are x = at t 2 , y = bt 2 and z = 0 , then its velocity at the instant t will be -

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TestHub · Accepted Answer

To find the velocity, we differentiate the position coordinates with respect to time.

Given: $x = a t^{2}$ , $y = b t^{2}$ , $z = 0$ .

Velocity components:

$v_{x} = \frac{d x}{d t} = \frac{d}{d t} (a t^{2}) = 2 a t$

$v_{y} = \frac{d y}{d t} = \frac{d}{d t} (b t^{2}) = 2 b t$

$v_{z} = \frac{d z}{d t} = \frac{d}{d t} (0) = 0$

The magnitude of the velocity is:

$v = v_{x}^{2} + v_{y}^{2} + v_{z}^{2}$

$v = (2 a t)^{2} + (2 b t)^{2} + (0)^{2}$

$v = 4 a^{2} t^{2} + 4 b^{2} t^{2}$

$v = 4 t^{2} (a^{2} + b^{2})$

$v = 2 t a^{2} + b^{2}$

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