---
title: "Oblique impact of ball on a wall"
subject: "Engineering Mechanics"
chapter: 4
key: "semester_3/engineering_mechanics:ball-oblique-impact-wall"
source: https://www.bctengineer.com/q/semester_3/engineering_mechanics/ball-oblique-impact-wall
site: BCT Engineering
---

# Oblique impact of ball on a wall

Engineering Mechanics · Chapter 4: Kinetics of Particles: Impulse and Momentum

- Asked in: year not given, 6 marks
- Video: [Oblique impact | Finding velocity and direction of ball after impact |Dynamics | coff.of restitution](https://www.youtube.com/watch?v=6tCTwi9GYgg) — Sudip Dhakal
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/engineering_mechanics/ball-oblique-impact-wall
- Whole chapter: https://www.bctengineer.com/md/semester_3/engineering_mechanics/chapter-4

## Question

A ball of mass 0.4 kg strikes a smooth fixed vertical wall with a speed of 20 m/s, its direction making an angle of 30° with the surface of the wall. The coefficient of restitution is 0.75. Find the speed and direction of the ball just after impact, the impulse on the wall and the average force if the contact lasts 0.01 s. Also find the kinetic energy lost.

## Short answer

Tangential component remains 17.32 m/s; normal component reverses with restitution: v_n = 0.75×10 = 7.5 m/s. Resultant speed is 18.87 m/s at 23.4° to the wall. Impulse on wall = 0.4×(1+0.75)×10 = 7.0 N·s; average force = 700 N; energy lost = 8.75 J.

## Answer

**Given:** $m = 0.4$ kg, $u = 20$ m/s at $30^\circ$ to the wall surface, $e = 0.75$. The wall is smooth, so the impulse acts along the normal only.

```text
   wall |
        |  normal (n)
        | <--
   -----+-------> tangent (t)
        |  \  incoming 30 deg to wall
```

Components of initial velocity:
- Normal (perpendicular to wall): $u_n = 20\sin30^\circ = 10.00$ m/s
- Tangential (along wall): $u_t = 20\cos30^\circ = 17.32$ m/s

#### Velocity after impact
- Tangential component is unchanged (no tangential impulse): $v_t = 17.32$ m/s
- Normal component reverses with restitution: $v_n = e\,u_n = 0.75 \times 10 = 7.50$ m/s

$$
v = \sqrt{v_n^2 + v_t^2} = \sqrt{7.5^2 + 17.32^2} = 18.87\ \text{m/s}
$$

Direction with the wall: $\tan\beta = v_n/v_t = 7.5/17.32$, so $\beta = 23.4^\circ$ (the ball leaves at $23.4^\circ$ to the wall, a flatter angle than $30^\circ$).

#### Impulse and average force
Impulse on the ball along the normal: $I = m(v_n + u_n) = m(1 + e)u_n = 0.4 \times 1.75 \times 10 = 7.00$ N·s. The impulse on the wall is equal and opposite.

$$
F_{avg} = \frac{I}{\Delta t} = \frac{7.00}{0.01} = 700\ \text{N}
$$

#### Kinetic energy lost

$$
\Delta T = \tfrac12(0.4)(20^2 - 356.25) = 8.75\ \text{J}
$$

**Answer: v = 18.87 m/s at 23.4° to the wall; impulse = 7.0 N·s; average force = 700 N; energy lost = 8.75 J.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. A ball hits a wall at 30 deg to the wall's surface. The normal component of velocity is:
   A. 20 cos30 = 17.32 m/s
   B. 20 sin30 = 10 m/s
   C. 20 tan30
   D. 20 m/s
   Answer: B. The angle is with the surface, so the normal component is 20 sin30 = 10 m/s. (Key point: Components)

2. After impact on a smooth wall:
   A. Tangential component is unchanged and normal component becomes e times u_n reversed
   B. Both components are multiplied by e
   C. Normal is unchanged and tangential is multiplied by e
   D. Both are unchanged
   Answer: A. No tangential impulse exists, so v_t = 17.32; v_n = 0.75 x 10 = 7.5 m/s. (Key point: After impact)

3. The impulse on the ball is:
   A. m(1 - e)u_n = 1.0 N s
   B. m(1 + e)u_n = 7.0 N s
   C. m u = 8 N s
   D. m e u_n = 3 N s
   Answer: B. The normal velocity changes from u_n to -e u_n, a change of (1 + e)u_n. (Key point: Impulse)

4. With contact lasting 0.01 s, the average force and energy lost are:
   A. 700 N and 8.75 J
   B. 70 N and 8.75 J
   C. 700 N and 0
   D. 7 N and 87.5 J
   Answer: A. F = 7/0.01 = 700 N; KE loss = 0.5(0.4)(400 - 356.25) = 8.75 J. (Key point: Average force and KE loss)

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