---
title: "Two balls impact e = 0.9, 30 and 50 kg"
subject: "Applied Mechanics (Dynamics)"
chapter: 2
key: "semester_3/applied_mechanics_dynamics:two-balls-e09-30-50kg"
source: https://www.bctengineer.com/q/semester_3/applied_mechanics_dynamics/two-balls-e09-30-50kg
site: BCT Engineering
---

# Two balls impact e = 0.9, 30 and 50 kg

Applied Mechanics (Dynamics) · Chapter 2: Kinetics of particles: Energy and Momentum Methods

- Asked in: 2072 Chaitra, 8 marks
- Video: [Dynamics 15-73| Two smooth disks A and B each have a mass of 0.5 kg.](https://www.youtube.com/watch?v=mxRWYKj0lTM) — Learning by Teaching
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/applied_mechanics_dynamics/two-balls-e09-30-50kg
- Whole chapter: https://www.bctengineer.com/md/semester_3/applied_mechanics_dynamics/chapter-2

## Question

The magnitude and direction of the velocities of two frictionless balls with the mass $m_A = 30$ kg and $m_B = 50$ kg before they strike each other are shown in figure below. Assume $e = 0.9$, determine the magnitude and direction of the velocity of each ball after the impact. [Figure: ball A (30 kg) moving right with $V_A = 30$ m/s along the horizontal; ball B (50 kg) with $V_B = 30$ m/s directed at $45^\circ$ below the horizontal, toward upper left.]

## Short answer

Smooth balls: tangential components unchanged. Using momentum and restitution (e=0.9) along horizontal line of impact, v'_A=30.8 m/s at 180° (rebounds), v'_B=26.1 m/s at 54.2° above +x.

## Answer

**Assumptions (figure description is ambiguous):** the line of impact is the horizontal x-axis. Ball A (30 kg) moves to the right at $30$ m/s. Ball B (50 kg) moves to the left and upward at $30$ m/s, $45^\circ$ to the line of impact. $e=0.9$, smooth balls.

#### Components before impact
$$
v_{Ax}=30,\ v_{Ay}=0;\qquad v_{Bx}=-30\cos45^\circ=-21.21,\ v_{By}=30\sin45^\circ=21.21\ \text{m/s}
$$

Tangential (y) components do not change: $v'_{Ay}=0$, $v'_{By}=21.21$ m/s.

#### Along the line of impact
Momentum:

$$
30(30)+50(-21.21)=30v'_{Ax}+50v'_{Bx}\ \Rightarrow\ 30v'_{Ax}+50v'_{Bx}=-160.7
$$

Restitution:

$$
v'_{Bx}-v'_{Ax}=0.9\,(30+21.21)=46.09
$$

Substituting $v'_{Bx}=v'_{Ax}+46.09$:

$$
80v'_{Ax}+2304.6=-160.7\ \Rightarrow\ v'_{Ax}=-30.82\ \text{m/s},\qquad v'_{Bx}=15.28\ \text{m/s}
$$

#### Final velocities
Ball A: $v'_A=30.82$ m/s, directed along the negative x-axis (it rebounds straight back, $180^\circ$).

Ball B:

$$
v'_B=\sqrt{15.28^{2}+21.21^{2}}=26.14\ \text{m/s},\qquad \theta=\tan^{-1}\frac{21.21}{15.28}=54.2^\circ\ \text{above the positive x-axis}
$$

**Answer: $v'_A=30.8$ m/s at $180^\circ$; $v'_B=26.1$ m/s at $54.2^\circ$ above the +x direction.** (If B instead moves down-left, the sign of B's y-component reverses and its direction becomes $54.2^\circ$ below +x.)

## Check yourself

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

1. What are ball B's components (30 m/s at 45 degrees to the line of impact)?
   A. -21.21 and 21.21 m/s
   B. -30 and 0
   C. 30 and 30
   D. -15 and 15
   Answer: A. 30 cos45 = 30 sin45 = 21.21; B moves in the -x direction. (Key point: Components before impact)

2. Which momentum equation applies along the line of impact?
   A. 30(30) + 50(-21.21) = 30 v'_Ax + 50 v'_Bx
   B. 30(30) + 50(21.21) = 30 v'_Ax + 50 v'_Bx
   C. 30 + 50 = v'_Ax + v'_Bx
   D. 30(30) = 50 v'_Bx
   Answer: A. B's x velocity is negative, so its momentum is -50(21.21). (Key point: Momentum equation)

3. What is v'_Bx - v'_Ax?
   A. 0.9 (30 + 21.21) = 46.09
   B. 0.9 (30 - 21.21)
   C. 30 + 21.21
   D. 0.9 x 30
   Answer: A. Relative approach speed is 30 - (-21.21) = 51.21 m/s. (Key point: Restitution equation)

4. What happens to ball A?
   A. It stops
   B. It rebounds straight back along -x at 30.8 m/s
   C. It continues forward at 30 m/s
   D. It moves upward
   Answer: B. A had no y velocity, which remains zero; v'_Ax = -30.82 m/s. (Key point: Ball A's result)

## Related in Applied Mechanics (Dynamics)

- [Two balls impact e = 0.84, energy lost](https://www.bctengineer.com/question.md?key=semester_3%2Fapplied_mechanics_dynamics%3Atwo-balls-e084-ke-lost) — Applied Mechanics (Dynamics), chapter 2
- [Two balls impact e = 0.67; work-energy principle](https://www.bctengineer.com/question.md?key=semester_3%2Fapplied_mechanics_dynamics%3Atwo-balls-e067-work-energy) — Applied Mechanics (Dynamics), chapter 2
- [Pool game, three balls, v0 = 5 m/s](https://www.bctengineer.com/question.md?key=semester_3%2Fapplied_mechanics_dynamics%3Apool-balls-three-elastic-5) — Applied Mechanics (Dynamics), chapter 2, asked 2×
- [Pool game, three balls, v0 = 4.57 m/s](https://www.bctengineer.com/question.md?key=semester_3%2Fapplied_mechanics_dynamics%3Apool-balls-three-elastic-4-57) — Applied Mechanics (Dynamics), chapter 2, asked 2×
- [Two slender rods released, angular velocity of AB](https://www.bctengineer.com/question.md?key=semester_3%2Fapplied_mechanics_dynamics%3Atwo-rods-released-beta-50) — Applied Mechanics (Dynamics), chapter 6
- [Cord pulled on disk, m = 30 kg, T = 200 N](https://www.bctengineer.com/question.md?key=semester_3%2Fapplied_mechanics_dynamics%3Acord-disk-m30-t200) — Applied Mechanics (Dynamics), chapter 5, asked 4×

## Related in other subjects

- [Frictionless piston 20 kg with H2O, five parts](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Africtionless-piston-20kg-h2o) — Fundamental of Thermodynamics & Heat Transfer, chapter 3
- [Oblique impact of ball on a wall](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_mechanics%3Aball-oblique-impact-wall) — Engineering Mechanics, chapter 4
- [Dip direction from strike N55°E, dip 30°](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_geology_i%3Adip-direction-from-strike-n55e) — Engineering Geology I, chapter 5
- [Resultant magnitude, direction, position about A](https://www.bctengineer.com/question.md?key=semester_1%2Fapplied_mechanics%3Aresultant-system-forces-about-a-beam) — Applied Mechanics, chapter 3