---
title: "Projectile exploding into two fragments (20 lb)"
subject: "Applied Mechanics (Dynamics)"
chapter: 3
key: "semester_3/applied_mechanics_dynamics:explosion-20lb-projectile"
source: https://www.bctengineer.com/q/semester_3/applied_mechanics_dynamics/explosion-20lb-projectile
site: BCT Engineering
---

# Projectile exploding into two fragments (20 lb)

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

- Asked in: 2076 Asoj, 6 marks; 2075 Asoj, 8 marks (asked 2 times)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/applied_mechanics_dynamics/explosion-20lb-projectile
- Whole chapter: https://www.bctengineer.com/md/semester_3/applied_mechanics_dynamics/chapter-3

## Question

A 20-lb projectile is moving with a velocity of 100 ft/s when it explodes into two fragments A and B, weighing 5 lb and 15 lb, 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

Using conservation of linear momentum in x and y: 20(100) = 5 v_A cos45° + 15 v_B cos30°, 0 = 5 v_A sin45° – 15 v_B sin30°. Solving yields v_A = 207.1 ft/s at 45° above the horizontal, v_B = 97.6 ft/s at 30° below the horizontal.

## Answer

The explosion forces are internal, so the total linear momentum is conserved. Weights are proportional to masses, so the factor $1/g$ cancels and the weights can be used directly.

```text
              A (5 lb)
               /  45 deg
 v0 =100 --->  *----------> x
               \  30 deg
              B (15 lb)
```

**Before:** the projectile moves horizontally at $100$ ft/s.

**x-direction:**
$$
20(100)=5v_A\cos45^\circ+15v_B\cos30^\circ
$$

$$
2000=3.5355v_A+12.990v_B\qquad(1)
$$

**y-direction** (initial vertical momentum is zero):
$$
0=5v_A\sin45^\circ-15v_B\sin30^\circ
$$

$$
3.5355v_A=7.5v_B\ \Rightarrow\ v_A=2.1213v_B\qquad(2)
$$

Substituting (2) in (1):

$$
2000=3.5355(2.1213v_B)+12.990v_B=20.490v_B
$$

$$
v_B=97.61\ \text{ft/s},\qquad v_A=2.1213(97.61)=207.06\ \text{ft/s}
$$

**Answer: $v_A=207.1$ ft/s at $45^\circ$ above the horizontal; $v_B=97.6$ ft/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. Why is total linear momentum conserved in the explosion?
   A. Gravity is absent
   B. The fragments have equal speeds
   C. Kinetic energy is conserved
   D. Explosion forces are internal
   Answer: D. Internal forces cancel in pairs, so total momentum just before and after is the same. (Key point: Conservation principle)

2. Why can weights be used in place of masses?
   A. Weights are in ft and masses are in lb
   B. The factor 1/g cancels from both sides
   C. g is zero in an explosion
   D. Weights equal momenta
   Answer: B. Weight is proportional to mass with the same 1/g factor on every term, so it cancels. (Key point: Use of weights)

3. What does the y-momentum equation state?
   A. 5 v_A cos45 = 15 v_B cos30
   B. 5 v_A sin45 + 15 v_B sin30 = 2000
   C. 0 = 5 v_A sin45 - 15 v_B sin30
   D. v_A = v_B
   Answer: C. The initial vertical momentum is zero and the fragments move on opposite sides, giving v_A = 2.1213 v_B. (Key point: y-direction equation)

4. What are the fragment speeds?
   A. v_A = 97.6 ft/s, v_B = 207.1 ft/s
   B. v_A = 207.1 ft/s, v_B = 97.6 ft/s
   C. v_A = 100 ft/s, v_B = 100 ft/s
   D. v_A = 141 ft/s, v_B = 115 ft/s
   Answer: B. Substituting v_A = 2.1213 v_B into 2000 = 3.5355 v_A + 12.99 v_B gives v_B = 97.6 ft/s and v_A = 207.1 ft/s; the light fragment is faster. (Key point: Final speeds)

## Close variants set in other exams

- [Define variable system; exploding projectile](https://www.bctengineer.com/question.md?key=semester_3%2Fapplied_mechanics_dynamics%3Avariable-system-define-explosion-20kg) — 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×