A simple but interesting problem

[dr_d_is_cool]

here is a relatively simple problem that is actually quite helpful and interesting. NOT HOMEWORK, i already have the solution, just a little problem for u guys.

a small sphere is released from rest, and, after falling a vertical distance of h, bounces on a smooth plane inclined at an angle theta to the horizontal. if the sphere loses no energy during the impact, why do its directions of motion immediately before and immediately after makeequal angles with the normal to th plane?

b)Find the distance, measured down the plane, between this impact and the next.

c) Find the ratio of the distances between the points at which the bouncing ball strikes the plane.


any questions don't hesitate to message me

 

[Nate810]

Solution: a) The reason why the directions of motion immediately before and immediately after make equal angles with the normal to the plane is because the kinetic energy of the sphere is conserved during the impact. Since the speed of the sphere is the same, the angles must be equal. b) The distance between the impact and the next is determined by the rebound angle, which is equal to the angle of incidence. Thus, the distance can be calculated as: d = h*tan(theta).c) The ratio of the distances between the points at which the bouncing ball strikes the plane can be calculated as: R = (h*tan(theta))/(h*tan(2*theta)).

 

[Moetasim]

I find this problem to be quite interesting and relevant. The phenomenon of a bouncing ball is a common occurrence in our daily lives, but understanding the mechanics behind it can be quite complex. It is great to see individuals taking an interest in these types of problems and seeking out solutions.

To address the first part of the problem, the fact that the sphere loses no energy during the impact is crucial in understanding why its directions of motion before and after the bounce make equal angles with the normal to the plane. This is due to the conservation of energy and the fact that the sphere is moving in a frictionless environment. This means that the total energy of the system (sphere and plane) remains constant, and the direction of motion of the sphere must change in order to maintain this conservation. Therefore, the angles before and after the bounce must be equal.

In terms of finding the distance between the first and second impacts, this can be calculated using basic trigonometry and the given information of the vertical distance fallen and the angle of inclination of the plane. This would also depend on the initial height at which the ball was released.

Finally, to find the ratio of distances between the two impacts, we would need to know the angle of incidence of the ball on the plane. This can be calculated using the angle of inclination of the plane and the angle of reflection, which would be equal to the angle of incidence. The ratio of distances can then be calculated using the law of sines.

Overall, this is a great problem that showcases the principles of energy conservation, motion, and basic trigonometry. It is important to continue exploring and understanding these concepts, as they have practical applications in many fields of science and technology. Thank you for sharing this problem and the opportunity to apply our knowledge in solving it.