---
title: "SHM amplitude from two speeds"
subject: "Engineering Physics"
chapter: 1
key: "semester_1/engineering_physics:shm-amplitude-from-speeds"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/shm-amplitude-from-speeds
site: BCT Engineering
---

# SHM amplitude from two speeds

Engineering Physics · Chapter 1: Oscillation

- Asked in: 2069 Asar, 5 marks
- Video: [How To Solve Simple Harmonic Motion Problems In Physics](https://www.youtube.com/watch?v=_Gnke2x3vT8) — The Organic Chemistry Tutor
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/shm-amplitude-from-speeds
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-1

## Question

A particle is moving with simple harmonic motion in a straight line. If it has a speed v1 when the displacement is x1 and speed v2 when the displacement is x2, then show that the amplitude of the motion is a = [(v2²x1² − v1²x2²)/(v2² − v1²)]^(1/2).

## Short answer

In SHM, v² = ω²(a² – x²). Substituting v1, x1 and v2, x2 and dividing eliminates ω, giving v1²/v2² = (a² – x1²)/(a² – x2²). Solving yields a = √((v2²x1² – v1²x2²)/(v2² – v1²)).

## Answer

For SHM, x = a sin(ωt + φ) and v = aω cos(ωt + φ). Eliminating t:
```text
v² = ω² (a² − x²)
```

#### Proof
1. At displacement x1 the speed is v1, and at x2 the speed is v2:
```text
v1² = ω² (a² − x1²)       ... (1)
v2² = ω² (a² − x2²)       ... (2)
```
2. Divide (1) by (2) to eliminate ω:
```text
v1² / v2² = (a² − x1²) / (a² − x2²)
v1² (a² − x2²) = v2² (a² − x1²)
```
3. Expand and collect terms in a²:
```text
v1² a² − v1² x2² = v2² a² − v2² x1²
a² (v1² − v2²) = v1² x2² − v2² x1²
a² (v2² − v1²) = v2² x1² − v1² x2²
```
4. Therefore
```text
a = [ (v2² x1² − v1² x2²) / (v2² − v1²) ]^(1/2)
```
which is the required result.

Also, from (1) − (2): ω² = (v1² − v2²)/(x2² − x1²), so the period T = 2π/ω can be found from the same measurements.

## Check yourself

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

1. In SHM v^2 =?
   A. w^2 (a^2 + x^2)
   B. w^2 x^2
   C. a^2 - x^2
   D. w^2 (a^2 - x^2)
   Answer: D. Eliminating t. (Key point: Speed relation)

2. How is w eliminated?
   A. Divide the two speed equations
   B. Add them
   C. Differentiate
   D. Set w = 1
   Answer: A. v1^2/v2^2 = (a^2 - x1^2)/(a^2 - x2^2). (Key point: Eliminating w)

3. a^2 =?
   A. (v1^2 x1^2 - v2^2 x2^2)/(v2^2 - v1^2)
   B. (v2^2 x1^2 - v1^2 x2^2)/(v2^2 - v1^2)
   C. (v2^2 x2^2 - v1^2 x1^2)/(v2^2 - v1^2)
   D. (v2^2 x1^2 + v1^2 x2^2)/(v2^2 + v1^2)
   Answer: B. Collect a^2 terms. (Key point: Result)

4. w^2 =?
   A. (v1^2 + v2^2)/(x2^2 + x1^2)
   B. (v2^2 - v1^2)/(x2^2 - x1^2)
   C. (v1^2 - v2^2)/(x2^2 - x1^2)
   D. v1/x1
   Answer: C. Subtracting the two equations. (Key point: Angular frequency)

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