A train is moving with a constant speed of `10m//s` in a circle of radius `16/pi`m. The plane of the circle lies in horizontal x-y plane. At time t = 0, train is at point P and moving in counter- clockwise direction. At this instant, a stone is thrown from the train with speed `10m//s` relative to train towards negative x-axis at an angle of `37^@` with vertical z-axis . Find
(a) the velocity of particle relative to train at the highest point of its
trajectory.
(b) the co-ordinates of points on the ground where it finally falls and that of the hightest point of its trajectory. (Take g `= 10 m//s^2, sin 37^@ = 3/5`

Answer

B, C

Explanation

At t = 0 `v_T = (10hatj)m//s`
`v_(ST) = 10 cos 37^@hat k - 10 sin 37^@hati = (8hatk-6hati)m//s`
`:. V_s = v_(ST) + v_(T) = (-6hati + 10hatj+8hatk)m//s`
(a) At highest point vertical component `(hatk)`of `v_s` will become zero. Hence, velocity of particle at highest point will become `(-6hati+10hatj)m//s`.
(b) Time of flight, `T = (2v_Z)/g = (2xx8)/10 = 1.6s`
`x = x_i + v_xT`
`= 16/pi -6 xx 1.6 = -4.5m`
`y = (10)(1.6) = 16 m and z = 0`
Therefore , coordinates of particle where it finally
lands on the ground are `(-4.5m, 16m, 0)`
At highest point, `t = T/2 = 0.8s`
`:. x = 16/pi - (6)(0.8) = 0.3m`
`y = (10)(0.8)= 8.0m`
and `z = (v_(Z)^(2))/(2g) = (8)^2/20 = 3.2m`
Therefore , coordinates at highest point are,
`(0.3 m, 8.0m, 3.2m)`.