First slide

Vectors

Question

Two billiard balls are rolling on a flat table. One has velocity components $v_{x} = 1 m / s, v_{y} = \sqrt{3} m / s$ and other has components , $v_{x}^{'} = 2 m / s and v_{y}^{'} = 2 m / s .$ . If both the balls start moving from the same point, the angle between their Paths is

Moderate

A

60°
B

45°
C

22.5°
D

15°

Solution

$\begin{array}{l} v_{1} = 1 i + \sqrt{3} j and v_{2} = 2 i + 2 j \\ v_{1} \cdot v_{2} = (i + \sqrt{3} j) \cdot (2 i + 2 j) = 2 + 2 \sqrt{3} \\ or v_{1} v_{2} \cos θ = 2 + 2 \sqrt{3} \end{array}$
$\begin{array}{l} \cos θ = \frac{1}{2} \times \frac{1}{\sqrt{2}} + \frac{\sqrt{3}}{2} \times \frac{1}{\sqrt{2}} \\ = \cos 60 \cos 45 + \sin 60 \sin 45 \\ = \cos (60 - 45) = \cos 15 \\ or θ = 15^{\circ} \end{array}$

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