EXPERIMENT 21 · PHYSICS LABORATORY

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. Explore a virtual apparatus, collect readings and analyse a graph.

CHANGE · OBSERVE · MEASURE

Your virtual workbench.

SWING HEIGHT OF RISE H—

MEASUREMENT STATUS

Ready for a virtual observation.

Pendulum bob mass M_p275.0 gSet Bob mass M_p (g)

Projectile ball mass m_b = 50.0 g (0.0500 kg); reference muzzle velocity v₀ = 5.000 m s⁻¹.

Adjust the apparatus, then take a reading.

Readings are generated by an ideal model. The diagram is schematic and is not to scale.

Simulation assumptions and limits

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.

Know your apparatus.

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.

FOLLOW THE METHOD

A reason behind every step.

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. Suggested independent settings: 150.0, 200.0, 275.0, 350.0, 450.0 g.

Apparatus Leveling and Equilibrium Index Alignment

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.

Why this step matters

Any tilt in the suspension axis causes the pendulum to swing out of plane, creating frictional rubbing against the ratchet track and systematically lowering the measured swing height.

Static Mass Metrology of Projectile and Pendulum Bob

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.

Why this step matters

Accurate mass values are essential to calculate total system mass M_total and transform post-collision momentum into projectile velocity.

Cocking the Spring Gun and Securing Firing Alignment

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.

Why this step matters

Consistent spring compression detents ensure repeatable projectile kinetic energy across successive trials.

Discharge Execution and Maximum Swing Height Measurement

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.

Why this step matters

The ratchet pawl eliminates visual estimation errors by physically arresting the indicator at the maximum elevation.

Systematic Pendulum Mass Variation via Accessory Weights

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.

Why this step matters

Symmetric loading preserves the lateral balance and center of percussion while systematically modulating the post-collision velocity V.

Coordinate Transformation, Regression, and Range Comparison

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.

Why this step matters

Linear regression over varying system masses filters random noise, tests momentum conservation, and yields an unbiased measurement of projectile muzzle velocity.

YOUR NOTEBOOK · 0 OBSERVATIONS

Turn measurements into a result.

See a complete example report ↗

Points are your observations. The fit uses SI x and y values before display scaling.

LINEAR FIT · y = mx + b

V = p₀ · (1 / M_total) ⇔ v₀ = S / m_b

#	Pendulum bob mass M_p (g)	Swing height of rise h (cm)	Inverse system mass 1/M (kg⁻¹)	Post-collision velocity V (m s⁻¹)	Remove

No readings yet. Take a virtual measurement and record it.

Calculate it yourself.

Your estimate of v₀ (m s⁻¹)

CHECK YOUR UNDERSTANDING

Explain what the measurements mean.

Why is kinetic energy not conserved during the collision between the projectile and the ballistic pendulum bob, whereas linear momentum is strictly conserved?

If an ordinary least-squares fit of post-collision velocity V (in m s⁻¹) versus inverse total mass 1/M (in kg⁻¹) yields a slope of S = 0.2503 kg m s⁻¹, and the projectile mass is m_b = 0.0500 kg, what is the initial muzzle velocity v₀?

A projectile of mass m_b is fired horizontally into a pendulum bob of mass M_p = 4 m_b. What percentage of the projectile's initial kinetic energy is converted into internal heat and deformation during the impact?

In the secondary verification using free projectile flight, if a ball fired horizontally from height H = 1.000 m lands at horizontal range R = 2.258 m under g = 9.807 m s⁻², what is the calculated muzzle velocity?