A car P is moving with a uniform speed `5sqrt3 m//s` towards a carriage of mass 9 kg at rest kept on the rails at a point B as shown in figure. The height AC is 120 m. Cannon balls of 1 kg are fired from the car with an initial velocity `100m//s` at an angle `30^@` with the horizontal. The first cannon hall hits the stationary carriage after a time`t_0` and sticks to it. Determine `t_0`.

At `t_0`, the second cannon ball is fired. Assume that the resistive force between the rails and the carriage is constant and ignore the vertical motion of the carriage throughout. If the second ball also hits and sticks to the carriage, what will be the horizontal velocity of the carriage just after the second impact?

Answer

`12s, 15.75 ms^(-1)`

Explanation

Consider the vertical motion of th ecannon ball
`:. S = ut + (1)/(2) at^(2) :. -120 = 50t_(0) - 5t_(0)^(2)`
`rArr 5t_(0)^(2) - 5t_(0) - 120 = 0 rArr t_(0)^(2) - 10 t_(0) - 24 = 0`
`:. t_(0) = -((-10) +- [sqrt(100) - 4(1)(-24))]/(2) = 12` or `-2`
The horizontal velocity of the cannon ball remains the same
`:. v_(x) = 100 cos30^(@) + 5sqrt3 = 55 sqrt3 m//s`

`:.` Apply conservation of linear momentum to the cannon ball-trolley system in horizontal direction. If m is the mass of cannon ball and M is the mass of the trolley then
`mv_(x) + M xx 0 = (m + M) V_(x) :. V_(x) = (mv_(x))/(m + M)`
where `v_(x)` is the velocity of the (cannon ball - trollwy) system
`V_(x) = (1 xx 55 sqrt3)/(1 + 9) = 5.5sqrt3 m//s`
The second ball was projected after 12 second
Horizontal distance covered by the car
`P = 12 xx 5sqrt3 = 60sqrt3m`
Since the second ball also struck the trollwy
`rArr` In time 12 seconds the trolley covers a distance of `60sqrt3`
For trolley in `12 sec`
From `s = ((u + v)/(2)) 60 sqrt3 = ((5.5sqrt3 + v)/(2)) (12) rArr v = 7.8 m//s`
To find the final velocity of the carriage after the second impact we again apply conservation of linear momentum in th ehorizontal direction
`mv_(x) + (M + m) 7.8 = (M + 2m) v_(f)`
`:. 1 xx 55 sqrt3 + (9 + 1) 7.8 = (9 + 2)v_(f)`
`rArr v_(f) = 15.75 m//s`