Oblique Collisions

In an oblique collision, the velocity of the object is not along the line of impact. This requires resolving velocities into components parallel and perpendicular to the surface or line of centres.

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1 Oblique Impact with a Smooth Wall

When a particle hits a smooth wall obliquely:

• The component parallel to the wall is unchanged (the wall is smooth, so no friction).
• The component perpendicular to the wall obeys Newton's law of restitution: the perpendicular rebound speed = e times the perpendicular approach speed.

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2 Worked Example 1

A ball hits a smooth wall at 30 degrees to the wall with speed 12 m s^(-1). The coefficient of restitution is 0.8. Find the speed and direction after the bounce.

Solution:

Perpendicular component (towards wall): 12 sin 30 = 6 m s^(-1)

Parallel component (along wall): 12 cos 30 = 6 sqrt(3) m s^(-1)

After bounce:

• Perpendicular: 0.8 * 6 = 4.8 m s^(-1) (away from wall)
• Parallel: 6 sqrt(3) m s^(-1) (unchanged)

Speed after = sqrt(4.8^2 + (6 sqrt(3))^2) = sqrt(23.04 + 108) = sqrt(131.04) approximately 11.4 m s^(-1)

Angle to wall: tan(beta) = 4.8 / (6 sqrt(3)) = 4.8/10.39 approximately 0.462

beta = arctan(0.462) approximately 24.8 degrees

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3 The tan beta = e tan alpha Rule

If the ball hits the wall at angle alpha to the wall surface and rebounds at angle beta:

tan beta = e tan alpha

This follows because:

• Perpendicular component changes by factor e
• Parallel component is unchanged

So tan(beta) / tan(alpha) = (e * perpendicular) / perpendicular * (parallel / parallel) = e.

When e = 1: beta = alpha (angle of incidence = angle of reflection, like light).

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4 Worked Example 2

A ball strikes a wall at 45 degrees with e = 0.5. Find the angle of rebound.

tan beta = 0.5 * tan 45 = 0.5 * 1 = 0.5

beta = arctan(0.5) approximately 26.6 degrees to the wall.

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5 Oblique Collision Between Two Spheres

When two smooth spheres collide obliquely:

• The line of centres connects the centres of the two spheres at the moment of impact.
• Newton's law of restitution applies only along the line of centres.
• The velocity components perpendicular to the line of centres are unchanged for each sphere (since the surfaces are smooth).

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