How is an oblique central impact solved, for a ball hitting a smooth surface or another ball?

In one line

Resolve velocities along and perpendicular to the line of impact; the perpendicular (tangential) components do not change, and the normal components follow momentum and restitution.

Explained simply

When a ball hits a smooth wall at an angle, the wall can only push it straight out, perpendicular to the wall. So the part of the velocity along the wall carries on unchanged, while the part into the wall reverses and shrinks according to e.

Technically, for smooth bodies in oblique central impact, along the tangent: each body's velocity component is unchanged. Along the line of impact (normal): use momentum conservation and \( e = (v_{2n} - v_{1n})/(u_{1n} - u_{2n}) \).

How it works

1. Draw the line of impact (n) and the tangent (t) at contact.
2. Resolve each initial velocity into n and t components.
3. t-direction: \( v_{1t} = u_{1t} \), \( v_{2t} = u_{2t} \).
4. n-direction: momentum \( m_1u_{1n} + m_2u_{2n} = m_1v_{1n} + m_2v_{2n} \) and the restitution equation.
5. For a fixed smooth surface: \( v_n = e\,u_n \) (reversed), \( v_t = u_t \).
6. Combine components to get final speed and direction.

Key formula

Fixed surface: \( v = \sqrt{(u\cos\theta)^2 + (e\,u\sin\theta)^2} \), rebound angle \( \tan\theta' = e\tan\theta \), where \( \theta \) is the incidence angle measured from the surface.

Quick example

A ball hits a smooth floor at 10 m/s, 30° to the floor, e = 0.5. Tangential: \( 10\cos30^\circ = 8.66 \) m/s unchanged. Normal: \( 10\sin30^\circ = 5 \) m/s becomes 2.5 m/s upward. Rebound speed \( = \sqrt{75 + 6.25} = 9.01 \) m/s at \( \tan^{-1}(2.5/8.66) = 16.1^\circ \) to the floor.

Common mistakes

• Applying e to the full velocity instead of the normal component.
• Changing the tangential components (they stay the same for smooth bodies).
• Measuring angles from the normal in one place and from the surface in another.

How MU asks it

6 to 8-mark numerical: 'A ball strikes a smooth wall/floor at ...; find the velocity and direction after impact', or two spheres colliding obliquely. Show the n-t axes on a sketch, a component table before and after, and the final magnitudes and angles.