The slope of wind screen of two cars are `alpha_(1)=30^(@)` and `a_(2)=15^(@)` respecively. At what ratio of `(v_(1))/(v_(2))` of the velocities of the cars will their drivers see the hall stones bounced back by the wind screen on their cars in vertical direction assume hall stones fall vertically downwards and collisions to be elastic?

Answer

3

Explanation

`v_("absolute")` in vertically downward `v_(He)` after collision vertically upwards since collison is clastic so velocity of hallisotnes w.r.t. car before and after collision will make equal angles in reference frame of car.

velocity of halistone relative to first car
`vecv_(HCl)=vecv_(H)-vecv_(C)=vecv-vecv_(1)`
`alpha=2beta=(pi)/(2)` and `tan alpha=(v)/(v_(1))`
Hence `tan alpha=tan((pi)/(2)-2beta)=cot 2beta`
or `(v)/(v_(1))=cot 2beta=cot 2alpha_(1)` as `beta=alpha_(1)`
Similarly for second car
`cot 2beta'=(v)/(v_(1))=cot 2alpha_(2)`
`therefore (v_(1))/(v_(2))=(cot 2alpha_(2))/(cot 2alpha_(1))=3`