A car initially travelling eastwards turns north by travelling in a quarter circular path of radius R metres at uniform speed as shown in figure. The car completes the turn in T second.
(a) What is the acceleration of the car when it is at B located at an angle of $37^{o}$ . Express your answers in terms of unit vectors $^i$ and $^j$
(b) The magnitude of car's average acceleration during T second period.

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Solution

$(a)$ At B, acceleration of car is centripetal and is along the center O of the circle.

Hence, unit vector along BO is $^a = - s i n 37^{o}^i + c o s 37^{o}^j$

$⟹^a = \frac{- 3 i + 4 j}{5}$

And, magnitude of acceleration is $a = \frac{v^{2}}{R} = \frac{1}{R} {(\frac{π R}{2 T})}^{2}$

$⟹ \to a = \frac{π^{2} R}{4 T^{2}} (\frac{- 3^i + 4^j}{5})$

$(b)$ $\to a = \frac{\to v - \to u}{T}$

$⟹ \to a = \frac{v^j - v^i}{T}$

$⟹ \to a = \frac{π R}{2 T^{2}} (^j -^i)$

$⟹ a = \frac{\sqrt{2} π R}{2 T^{2}}$

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