---
title: "Define variable system; exploding projectile"
subject: "Applied Mechanics (Dynamics)"
chapter: 3
key: "semester_3/applied_mechanics_dynamics:variable-system-define-explosion-20kg"
source: https://www.bctengineer.com/q/semester_3/applied_mechanics_dynamics/variable-system-define-explosion-20kg
site: BCT Engineering
---

# Define variable system; exploding projectile

Applied Mechanics (Dynamics) · Chapter 3: System of particles

- Asked in: 2081 Bhadra, 2+6 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/applied_mechanics_dynamics/variable-system-define-explosion-20kg
- Whole chapter: https://www.bctengineer.com/md/semester_3/applied_mechanics_dynamics/chapter-3

## Question

Define variable system of particles. A 20 kg projectile is moving with a velocity of 30 m/s when it explodes into two fragments A and B, weighing 5 kg and 15 kg, respectively. Knowing that immediately after the explosion, fragments A and B travel in directions defined respectively by $\theta_A = 45^\circ$ and $\theta_B = 30^\circ$, determine the velocity of each fragment. [Figure: 20 lb projectile moving right at $v_0 = 100$ ft/s; after explosion, 5 lb fragment A moves at $\theta_A = 45^\circ$ above the horizontal and 15 lb fragment B at $\theta_B = 30^\circ$ below the horizontal.]

## Short answer

A variable system of particles gains or loses mass continuously. Momentum conservation for the explosion: 20(30) = 5 v_A cos45° + 15 v_B cos30°, 0 = 5 v_A sin45° – 15 v_B sin30°. Results: v_A = 62.12 m/s at 45° above horizontal, v_B = 29.28 m/s at 30° below.

## Answer

#### Variable system of particles
A **variable system of particles** is one whose total mass changes with time because particles continuously join it or leave it, such as a rocket ejecting gas, a conveyor belt loaded with sand, or a chain being lifted from a heap. Its equation of motion is $\sum\vec F=m\vec a-\dfrac{dm}{dt}\vec u_{rel}$.

#### Exploding projectile
Internal forces of the explosion do not change the total momentum. Masses: $m=20$ kg, $m_A=5$ kg, $m_B=15$ kg, $v_0=30$ m/s (the numbers given in the question text; the figure shows the same arrangement in lb and ft/s).

```text
              A (5 kg)
               /  45 deg
 v0 = 30 --->  *----------> x
               \  30 deg
              B (15 kg)
```

**x-direction:**
$$
20(30)=5v_A\cos45^\circ+15v_B\cos30^\circ
$$

$$
600=3.5355v_A+12.990v_B\qquad(1)
$$

**y-direction:**
$$
0=5v_A\sin45^\circ-15v_B\sin30^\circ\ \Rightarrow\ v_A=2.1213v_B\qquad(2)
$$

Substituting (2) in (1):

$$
600=7.5v_B+12.990v_B=20.490v_B\ \Rightarrow\ v_B=29.28\ \text{m/s}
$$

$$
v_A=2.1213(29.28)=62.12\ \text{m/s}
$$

**Answer: $v_A=62.12$ m/s at $45^\circ$ above the horizontal; $v_B=29.28$ m/s at $30^\circ$ below the horizontal.**

## Check yourself

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

1. Which is an example of a variable system of particles?
   A. A pendulum bob
   B. A block sliding on a floor
   C. A satellite in orbit
   D. A rocket ejecting gas
   Answer: D. A variable system's mass changes with time as particles join or leave, like a rocket, sand-loaded conveyor or chain lifted from a heap. (Key point: Definition)

2. Why do the explosion forces not change the total momentum?
   A. They are internal forces
   B. They act only vertically
   C. The fragments are equal in mass
   D. They are conservative
   Answer: A. Internal forces occur in equal and opposite pairs, so they cancel in the total. (Key point: Momentum conservation)

3. What does the y-direction equation give?
   A. v_A = 0.4714 v_B
   B. v_A = v_B
   C. v_A = 2.1213 v_B
   D. v_A = 2.5 v_B
   Answer: C. 5 v_A sin45 = 15 v_B sin30 gives 3.5355 v_A = 7.5 v_B, so v_A = 2.1213 v_B. (Key point: y equation relation)

4. What are the fragment speeds?
   A. v_A = 29.28 m/s, v_B = 62.12 m/s
   B. v_A = 62.12 m/s, v_B = 29.28 m/s
   C. v_A = 30 m/s, v_B = 30 m/s
   D. v_A = 97.6 m/s, v_B = 207 m/s
   Answer: B. 600 = 20.49 v_B gives v_B = 29.28 m/s and v_A = 62.12 m/s. (Key point: Final speeds)

## Close variants set in other exams

- [Projectile exploding into two fragments (20 lb)](https://www.bctengineer.com/question.md?key=semester_3%2Fapplied_mechanics_dynamics%3Aexplosion-20lb-projectile) — Applied Mechanics (Dynamics), chapter 3, asked 4×
- [Define angular momentum; exploding 10 kg projectile](https://www.bctengineer.com/question.md?key=semester_3%2Fapplied_mechanics_dynamics%3Aangular-momentum-define-explosion-10kg) — Applied Mechanics (Dynamics), chapter 3, asked 4×