A classical mechanics problem by Aakhyat Singh

Classical Mechanics Level 2

A ball falls on a fixed inclined plane of inclination θ \theta from a height of h h above the point of impact, making a perfectly elastic collision.

If it hits the plane again at a distance of x h sin ⁡ θ xh \sin \theta from the point of the first impact, find x x . Assume the inclined surface to be frictionless.


The answer is 8.

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Aakhyat Singh by Aakhyat Singh , 20, India

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1 solution

Sardor Yakupov
Aug 23, 2017

It is easier to solve this problem with respect to a plane. Then:

g x = g sin ⁡ θ g y = g cos ⁡ θ v x = 2 g h sin ⁡ θ v y = 2 g h cos ⁡ θ t = 2 v y g y = 2 2 g h cos ⁡ θ g cos ⁡ θ = 2 2 h g S = v x t + g x t 2 2 = 4 h sin ⁡ θ + 4 h sin ⁡ θ = 8 h sin ⁡ θ x = 8 { g }_{ x }=g\sin { \theta } \\ { g }_{ y }=g\cos { \theta } \\ \\ { v }_{ x }=\sqrt { 2gh } \sin { \theta } \\ { v }_{ y }=\sqrt { 2gh } \cos { \theta } \\ \\ t=\frac { 2{ v }_{ y } }{ { g }_{ y } } =\frac { 2\sqrt { 2gh } \cos { \theta } }{ g\cos { \theta } } =2\sqrt { \frac { 2h }{ g } } \\ S={ v }_{ x }t+\frac { { g }_{ x }{ t }^{ 2 } }{ 2 } =4h\sin { \theta } +4h\sin { \theta } =8h\sin { \theta } \\ \\ x=8

6 Helpful 0 Interesting 0 Brilliant 0 Confused

why is the x-component of g with respect to sin and not cos? and same for y component of g. why is it with respect to cos? is there a switch of right triangles?

Matthew Agona - 3 years, 9 months ago

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Sorry for my paintning skills. https://s8.hostingkartinok.com/uploads/images/2017/09/bca8b68db0fa9a57fd4b10a4969566fe.png

Sardor Yakupov - 3 years, 9 months ago

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hey thanks for the graphic!

Matthew Agona - 3 years, 9 months ago

i don't understand why time = 2vy / gy

Breno Lemos - 3 years, 9 months ago

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At the time t = v y g y t=\frac { { v }_{ y } }{ { g }_{ y } } v y = 0 { v }_{ y }=0 and a ball is on top of his trajectory (with respect to plane). A ball will fall from this point same time, so t u p = t d o w n = v y g y { t }_{ up }={ t }_{ down }=\frac { { v }_{ y } }{ { g }_{ y } }

Full time is the sum of this two times. t = 2 v y g y t=\frac { 2{ v }_{ y } }{ { g }_{ y } }

Sardor Yakupov - 3 years, 8 months ago

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Thank you so much

Breno Lemos - 3 years, 8 months ago

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