A particle moves according to the law x = r cos ⁡ π t 2 . The distance covered by it in the time interval between t = 0 and t = 3 s is n r . What is the value of n ?

Step-by-step solution

Given, x = r cos ⁡ π t 2 ⇒ v = d x d t = - π 2 r sin ⁡ π t 2 Graphically it can be represented as shown in figure. Distance covered by the particle = Magnitude of area of x - t curve between 0 to 2 s and 2 to 3 s = ∫ 0 2 v d t + ∫ 2 3 v d t = - ∫ 0 2 π 2 r sin ⁡ π 2 t d t + ∫ 2 3 - π 2 r sin ⁡ π 2 t d t = - π 2 r ∫ 0 2 sin ⁡ π t 2 d t + - π 2 r ∫ 2 3 sin ⁡ π 2 t d t = π r 2 cos ⁡ π t 2 π 2 0 2 + π r 2 cos ⁡ π 2 t π 2 2 3 = r cos ⁡ π - cos ⁡ 0 + r cos ⁡ 3 π 2 - cos ⁡ π = r - 1 - 1 + r 0 - - 1 = 2 r + r = 3 r