Two identical billiard balls strike a rigid wall with the same speed but at different angles, and get reflected without any change in speed, as shown in Fig. 5.6. What is (i) the direction of the force on the wall due to each ball? (ii) the ratio of the magnitudes of impulses imparted to the balls by the wall ?
Let the angle of incidence of ball 1 onto the wall be = A
Let the angle of incidence of the ball 2 be = B
Let the speed of the balls just before striking the wall = v
The velocity vectors (incident and reflected) are resolved along the normal to the wall and along the wall.
For the change in velocity of ball 1 = vf - vi
= (v cos A j + v sin A i) - (v sin A i - v cos A j)
= 2 v cos A j
Impulse = change in the momentum of ball 1 = 2 m v cos A j
So the direction of force = direction of momentum = normal to the wall, coming out of the wall.
ratio of impulses = (2 m v cos A) / (2 m v Cos B) = cosA / cos B
= ratio of cosines of angles of incidence.
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