Example Data Table
| Projectile mass (g) | Pendulum mass (kg) | Length (m) | Max angle (deg) | Computed speed (m/s) |
|---|---|---|---|---|
| 10 | 2.0 | 0.75 | 12 | ≈ 107 |
| 8 | 1.5 | 0.60 | 9 | ≈ 89 |
| 15 | 3.0 | 1.00 | 7 | ≈ 63 |
These rows are illustrative. Real results depend on measurement quality and energy losses.
Formula Used
- Swing height from angle: h = L(1 − cos θ)
- From energy during swing: ½(M+m)V² = (M+m)gh → V = √(2gh)
- From momentum in the collision: m v = (M+m) V → v = ((M+m)/m) · V
- Optional displacement relation: x = L sin θ → θ = asin(x/L)
Here m is projectile mass, M is pendulum mass, L is pendulum length, g is gravity, V is speed right after impact, and v is projectile speed before impact.
How to Use This Calculator
- Choose a mode: solve projectile speed, or predict the swing.
- Enter projectile mass, pendulum mass, and pendulum length.
- Keep the length measured to the pendulum’s center of mass.
- For speed-from-swing, select angle, rise, or displacement.
- Press Calculate to view results above the form.
- Use CSV/PDF buttons to export the computed summary.
For best accuracy, repeat trials and average the swing measurement.
Ballistic Pendulum Overview
A ballistic pendulum estimates a projectile’s incoming speed by capturing it in a hanging block and measuring how far the combined mass swings. The collision is treated as perfectly inelastic, so momentum is conserved during impact, then mechanical energy is conserved during the swing up to the peak.
Typical Input Ranges
In classroom setups, projectile masses often fall between 5–20 g, while the pendulum block is commonly 1–5 kg. A practical length L is 0.5–1.2 m; longer lengths produce larger, easier-to-read angles. For many demonstrations, peak angles are kept around 3–20°, which corresponds to rises like 2–60 mm when L is near 0.75 m.
Why Mass Ratio Matters
The speed after impact V is reduced by the factor m/(M+m). If the projectile is 10 g and the block is 2.0 kg, only about 0.5% of the projectile’s speed remains in V, so the swing may be subtle. Using a lighter block increases swing size, but you must keep the catcher rigid for repeatability.
Turning Swing Into Height
This calculator lets you enter the swing as an angle θ, a vertical rise h, or a horizontal displacement x. The geometric link is h = L(1 − cosθ). For small angles, cosθ ≈ 1 − θ²/2, so h ≈ Lθ²/2; that is why a reading error in θ can change h noticeably.
Energy Loss and What It Means
Kinetic energy is not conserved during impact because the projectile embeds, deforms, and produces heat and sound. The tool reports energy lost as the difference between projectile kinetic energy (½mv²) and post-impact kinetic energy (½(M+m)V²). A large loss is normal and does not invalidate the momentum method.
Improving Data Quality
Use a consistent release point, measure L to the combined center of mass, and repeat at least 5 trials. If you read angles, use a simple pointer and protractor; a 1° error at 10° can shift the speed by several percent. If you measure displacement, keep x/L ≤ 0.3 to avoid extreme angles.
Reporting Results Clearly
Record inputs with units, include v, V, θ, and h, and note your gravity assumption (9.80665 m/s² by default). Export CSV for lab sheets and PDF for attachments, then compare trials using averages and deviation.
FAQs
1) What does a ballistic pendulum calculate?
It estimates the projectile’s incoming speed by measuring the swing of a block that captures the projectile. The method combines momentum conservation during impact with energy conservation during the upward swing.
2) Which swing input should I use: angle, rise, or displacement?
Use angle if you can read a protractor cleanly. Use rise if you can measure the vertical lift directly. Use displacement when you have a ruler at the arc; keep x/L modest for best accuracy.
3) Why does the calculator show energy loss?
The collision is inelastic, so some kinetic energy becomes heat, sound, and deformation. Momentum still applies during impact, so the speed estimate remains valid even when the loss value is large.
4) How can I improve accuracy?
Measure length to the combined center of mass, keep the pivot low-friction, and repeat multiple trials. Average the swing measurement, and avoid very small angles where reading error dominates the height calculation.
5) What if my angle is very large?
Large angles increase geometric nonlinearity and make air drag and pivot friction more influential. If θ approaches 60° or more, treat results as approximate and consider using a different method or applying detailed corrections.
6) Can I use this for different locations or planets?
Yes. Enter the local gravitational acceleration in m/s². A smaller g increases the predicted swing height for the same post-impact speed, so using the correct value helps when comparing experiments in different places.