2D Motion of Sphere - Inclined Plane

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SUMMARY

The discussion focuses on calculating the distance down an inclined plane that a small sphere travels after being dropped from a height 'h'. The key conclusion is that the distance between the first impact and the next impact point is 8hSinθ. The solution involves using a rotated reference frame to analyze the motion, applying the equations of motion along the inclined plane and the vertical direction. The equations used include displacement along the inclined plane and vertical displacement, leading to the derivation of the distance formula.

PREREQUISITES
  • Understanding of basic physics concepts such as motion and gravity.
  • Familiarity with inclined planes and their effects on motion.
  • Knowledge of kinematic equations for linear motion.
  • Ability to apply trigonometric functions, specifically sine and cosine.
NEXT STEPS
  • Study the derivation of kinematic equations for inclined planes.
  • Learn about energy conservation principles in elastic and inelastic collisions.
  • Explore the concept of rotated reference frames in physics.
  • Investigate the effects of different angles of inclination on projectile motion.
USEFUL FOR

Students studying physics, particularly those focusing on mechanics and motion, as well as educators looking for examples of inclined plane problems in kinematics.

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Homework Statement



A small sphere is dropped from rest at a height 'h' above a plane inclined at an angle θ to the horizontal ( < 90degrees ). Given that the sphere loses no energy on impact, show that the distance down the plane between this impact and the next is 8hSinθ.


Homework Equations





The Attempt at a Solution



Presumably, we need to consider a rotated reference frame oriented to apply to the given inclined plane. The component along the plane will have a Sinθ component, but I can't really put my finger on how to solve this.
 
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After the first impact, the ball will recoil with velocity v making an angle θ with the perpendicular to the inclined plane.
If A is the first point of impact and P is the next point of impact, then distplacement along inclined plane is
AP = vsinθ*t + 1/2*gsinθ*t^2...(1)
Along the vertical direction to the inclined plane, the displacement is zero. So
0 = vcosθ*t - 1/2*gcosθ*t^2 ...(2)
Find the value of t from the second equation and substitute in eq(1) to find AP.
 

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