First slide

Motion in two and three dimensions

Question

A particle starts from rest from origin and is moving in XY-plane, its component of acceleration in the direction of instantaneous velocity is $a_{0} = \vec{a} \cdot \hat{v}$ . where a is a constant vector of magnitude 2 ms^-2 directed along Y-axis. Find its speed in ms^-1 at y =1 m.

Difficult

Solution

$a_{0} = | \vec{a} ∥ \hat{v} | \cos θ$
$Or v \frac{d v}{d s} = a \cos θ or v d v = a d s \cos θ$
$\begin{array}{l} ∵ s = x \hat{i} + y \hat{j} or d s = d x \hat{i} + d y \hat{j} \\ ∴ v d v = a \cdot d s = a \hat{j} \cdot (d x \hat{i} + d y \hat{j}) \end{array}$
$\begin{array}{l} or \int_{0}^{v} v d v = a d y = 2 \int_{0}^{y} d y (∵ a = 2 {ms}^{- 2}) \\ or \frac{v^{2}}{2} = 2 y \\ or v = \sqrt{4 y} = \sqrt{4 \times 1} = 2 {ms}^{- 1} \end{array}$

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