Ballistic pendulum and momentum conservation lab report

1. Title and aim

Conservation of momentum and kinetic energy in an inelastic collision using a ballistic pendulum. Determine the muzzle velocity v₀ and initial momentum p₀ of a high-speed projectile fired from a spring gun into a ballistic pendulum bob, verifying the Law of Conservation of Linear Momentum during a completely inelastic collision across varying pendulum masses M_p, and comparing the momentum-derived velocity against free-flight projectile kinematics.

2. Abstract

Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of Projectile muzzle velocity of 100.1298 m s⁻¹, differing by 0.130% from the model value. The reported scatter comes from scale rounding and any stated model approximation, not measured laboratory noise.

3. Apparatus and fixed settings

• Precision Spring Gun with Triple Compression Detents: Fires a 50.0 g ground steel sphere horizontally with highly repeatable spring compression and negligible muzzle dispersion.
• Ballistic Pendulum Bob with Internal Spring Catch: A hollow cylindrical aluminum/brass pendulum bob fitted with an internal spring-loaded catch to trap the ball at its center of percussion, plus lateral studs for accessory masses.
• Toothed Elevation Arc and Anti-Reversal Ratchet Pawl: A curved aluminum protractor/metric arc that catches a lightweight indicator pawl at the apex of the pendulum's swing, preserving the maximum height reading h.
• Accessory Brass Loading Cylinders and Ground Metrology Scale: Pair of matched cylindrical brass masses (50 g, 75 g, 100 g) mounted symmetrically on the bob, and a vertical steel rule reading to 0.1 mm.
• Projectile ball mass m_b = 50.0 g (0.0500 kg); reference muzzle velocity v₀ = 5.000 m s⁻¹.
• Ballistic pendulum suspension length L = 0.400 m; ratchet height scale resolution 0.1 mm (0.01 cm).
• Local gravitational acceleration g = 9.80665 m s⁻².

4. Theory and graph choice

The ballistic pendulum, invented in 1742 by English mathematician and military engineer Benjamin Robins, stands as one of the most ingenious classical experimental apparatuses in physics. Prior to Robins' work, the velocities of high-speed projectiles such as cannonballs and musket balls could not be measured because their flight durations over laboratory distances were far too brief for mechanical clocks. Robins recognized that while mechanical kinetic energy is largely dissipated during a violent collision, the total linear momentum of an isolated system is strictly conserved. In a ballistic pendulum experiment, a projectile of mass m_b traveling at an unknown high velocity v₀ strikes and lodges permanently inside a massive suspended pendulum bob of mass M_p. Because the collision is completely inelastic, the combined mass M_total = m_b + M_p moves immediately after impact with a common initial swing velocity V. According to the Law of Conservation of Linear Momentum, the initial momentum of the flying ball equals the momentum of the combined system immediately following the impact: p₀ = m_b · v₀ = (m_b + M_p) · V. Following the impact, the pendulum swings upward along a circular arc against the restoring force of gravity. During this smooth upward swing, mechanical energy is conserved: the post-collision kinetic energy 1/2 M_total V² is converted entirely into gravitational potential energy M_total g h at the maximum height of rise h, establishing that V = sqrt(2 g h). By rearranging the momentum conservation equation as V = p₀ · (1 / M_total), we obtain a linear relationship between post-collision velocity V and the reciprocal of the total system mass 1/M_total. By systematically altering the pendulum bob mass M_p using calibrated accessory brass loading cylinders, students obtain a linear plot whose slope S directly yields the initial momentum p₀ = m_b · v₀, allowing the projectile's muzzle velocity to be determined with high precision as v₀ = S / m_b. Furthermore, the ballistic pendulum result can be independently validated by removing the pendulum and allowing the projectile to fly freely from the tabletop, determining v₀ from the horizontal range R and vertical drop distance H.

The calculation fits y = mx + b in SI units. The graph may display scaled axes for readability; those display factors are removed before calculating the physical result. The gradient has units kg m s⁻¹.

5. Method

1. Place the ballistic pendulum on a stable, vibration-free laboratory bench. Adjust the leveling screws at the cast-iron base until the pendulum hangs plumb and swings freely in a single vertical plane without lateral wobble. Zero the indicator pawl against the base of the toothed ratchet arc.
2. Weigh the steel projectile ball m_b (50.0 g) and the unloaded pendulum bob M_p (150.0 g) on a digital analytical balance. Record the baseline mass values. Measure the effective distance from the pivot knife-edge to the center of gravity of the combined pendulum-ball assembly.
3. Insert the steel ball into the spring gun barrel. Using the ramrod, compress the spring to the medium compression detent until the sear engages with a distinct click. Verify that the axis of the barrel points directly into the receiving mouth of the pendulum bob.
4. Trip the trigger release smoothly. The projectile discharges into the bob, where the internal spring catch traps it completely. As the pendulum swings upward, the indicator pawl advances along the toothed arc and locks at the peak. Record the height of rise h (in cm) from the vernier scale to 0.01 cm.
5. Attach matched brass loading cylinders symmetrically to the lateral studs of the pendulum bob, setting total bob mass M_p to 200.0 g, 275.0 g, 350.0 g, and 450.0 g. Fire the projectile three times at each mass setting, recording the mean swing height h for each configuration.
6. For each mass station, compute total mass M = (M_p + m_b)/1000, inverse mass 1/M, and post-collision velocity V = sqrt(2gh). Plot V versus 1/M, perform an ordinary least-squares linear fit to find slope S = p₀, and compute muzzle velocity v₀ = S / m_b. Compare with horizontal range data.

6. Observations and calculation table

All observations use fixed model settings. Values are rounded only at the instrument reading stage; further calculations use full precision. Repeated physical trials are not claimed.

Download the sample observations (CSV) ↓

7. Worked calculation for one observation

8. Graph, gradient and final calculation

Dots show the sample observations. The line is the ordinary least-squares fit.

9. Uncertainty and comparison

Percentage difference = |estimate − model value| / |model value| × 100 = 0.1298%. Percentage difference is not experimental uncertainty.

For n = 5 observations, Σresidual² = 5.1310e-07; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 1.7234e-04. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 6.0056e-04.

Propagating only the relevant fit coefficient gives a fit-only standard error of 0.06894 m s⁻¹. This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.

Ratchet tooth spacing resolution (±0.5 mm in rise height), friction work dissipated by the ratchet pawl during swing, suspension string mass and rotational inertia of the pendulum arm, air resistance during the swing arc, and imperfectly horizontal firing alignment contribute uncertainty. The OLS regression standard error reflects random scatter across pendulum mass settings, but does not include shared projectile mass balance calibration errors or mechanical spring fatigue in the launcher.

10. Result and discussion

The observations follow the stated model relationship. Check whether the intercept and residual pattern suggest a systematic offset or a missing approximation. Agreement with the model does not independently verify its input constants.

11. Precautions and limitations

• Verify that the internal spring catch of the pendulum bob completely retains the projectile on every trial so that the collision is strictly completely inelastic.
• Ensure the ratchet indicator pawl glides freely along the toothed arc without sticking or applying excessive downward friction on the swinging bob.
• Align the spring gun barrel horizontally and confirm it points directly into the receiving bore of the pendulum bob at equilibrium.
• Mount accessory brass weights symmetrically on both lateral studs of the bob to maintain dynamic balance and prevent torsional oscillations.

This virtual experiment assumes a point-mass pendulum suspension with zero pivot friction, negligible ratchet pawl drag, and zero aerodynamic resistance. It assumes the projectile's trajectory aligns perfectly with the center of percussion. Generated data reflect an ideal inelastic collision; physical experiments will exhibit small random variations from spring launcher release vibrations. Theoretical post-collision velocity: V = sqrt(2gh). Linearized relation: V = p₀ · (1 / M_total), where p₀ = m_b · v₀. The ordinary least-squares regression extracts the slope S = p₀, from which projectile velocity is recovered as v₀ = S / m_b. Reference projectile muzzle velocity v₀ = 5.000 m s⁻¹.

12. Conclusion

The worked example illustrates how to obtain Projectile muzzle velocity from measurements, graph variables and a fitted coefficient. The estimate is 100.1298 m s⁻¹. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.