- #1

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- Homework Statement
- An object with mass m free fall and collides with an inclined plane that cannot move with velocity V. The angle between the vector of final velocity and a line that is perpendicular to the surface?

- Relevant Equations
- Newton third law

impulse-momentum theorem

In my textbook, it is stated that "if an object elastically hit an frictionless inclined surface with angle between the vector of initial velocity and an imaginary line that is perpendicular to the surface ##\alpha##,then the angle between the line and final velocity vector will also be ##\alpha##. Fig.1 illustrates this statement

However, I'm not sure that this is true,so I'm going to prove it.

Note:

1. At X axis, the final velocity is marked as ##V'_x## ,and ##V_x##means initial velocity

2. Fig.2 illustrates the force that works on the object

M's impulse at X-axis :

$$−N_x\,dt=dP_{x_M} $$$$∫−N_x\,dt=ΔP_{x_M}$$

since the velocity of the plane does not change, then ##\Delta P_{x_M}## is 0. As a result :$$∫N_x\,dt=0...(1)$$

Now, for the impulse works on ##m## at X-axis : $$N_x\,dt=dP_{x_m}$$$$ ∫N_x\,dt=ΔP_{x_m}...(2)$$ Substitute equation 1 into equation 2

$$0=m(V'_{x_m} − V_{x_m})$$ Since the object free fall,the initial velocity of ##m## in X axis is 0$$V'_{x_m}=0$$

This breaks my textbook's statement ,and this does not make any sense because as in fig.1 , we know exactly that ##m## has ##V'_x##

It can be concluded that my method is forbidden.

So,my question is : Is there any law that my method breaks?

However, I'm not sure that this is true,so I'm going to prove it.

Note:

1. At X axis, the final velocity is marked as ##V'_x## ,and ##V_x##means initial velocity

2. Fig.2 illustrates the force that works on the object

M's impulse at X-axis :

$$−N_x\,dt=dP_{x_M} $$$$∫−N_x\,dt=ΔP_{x_M}$$

since the velocity of the plane does not change, then ##\Delta P_{x_M}## is 0. As a result :$$∫N_x\,dt=0...(1)$$

Now, for the impulse works on ##m## at X-axis : $$N_x\,dt=dP_{x_m}$$$$ ∫N_x\,dt=ΔP_{x_m}...(2)$$ Substitute equation 1 into equation 2

$$0=m(V'_{x_m} − V_{x_m})$$ Since the object free fall,the initial velocity of ##m## in X axis is 0$$V'_{x_m}=0$$

This breaks my textbook's statement ,and this does not make any sense because as in fig.1 , we know exactly that ##m## has ##V'_x##

It can be concluded that my method is forbidden.

So,my question is : Is there any law that my method breaks?