Ballistic Pendulum Experiment Overview and Key Concepts

Goals of the Lab

  • To apply the conservation of energy and momentum to determine the velocity of a projectile in a real-world problem.

Key Concepts

  • Energy Conversion:

    • The conversion of an object’s Kinetic Energy into Gravitational Potential Energy.

    • Associated formulas to review:

    • Kinetic Energy (KE): KE=12mv2KE = \frac{1}{2} mv^2

    • Potential Energy (PE): PE=mghPE = mgh

  • Momentum:

    • Momentum is always conserved during collisions.

    • For the ballistic pendulum experiment:

    • Initial momentum of the projectile (waterball) = mvmv

    • Final momentum after collision = (m+M)V(m+M)V

    • Conservation equation: mv=(m+M)Vmv = (m + M)V

  • Uncertainty:

    • Many physical quantities must be calculated from measurements, which can introduce uncertainty.

    • Strategies to minimize uncertainty include careful measurement and propagation of errors, typically calculated as:

    • If q≡q(x)q ≡ q(x), then δq=dqdxxδx\delta q = \frac{dq}{dx} x \delta x

    • For sums: q=x+y−zq = x + y - z, then δq=δx2+δy2+δz2\delta q = \delta x^2 + \delta y^2 + \delta z^2

Experiment Overview

  • Measurement Approach:

    • History: Traditionally, the speed of a bullet was measured by shooting it into a block of wood and assessing the swing of the block.

    • Current Experiment: Measure the velocity of a waterball launched from a waterball launcher into a suspended bottle.

Basic Theory for Derivation

  • Use of Conservation Laws:

    1. Momentum conservation (inelastic collision)

    2. Energy conservation as the bottle swings and rises to height hh.

  • Formulas:

    • From momentum conservation:

    • Solve for projectile velocity:
      v=(m+M)Vmv = \frac{(m + M)V}{m}

    • Kinetic and Potential Energy relationship:

    • Equate energies: KE=PEKE = PE

    • Substituting gives:
      12(m+M)V2=(m+M)gh\frac{1}{2}(m + M)V^2 = (m + M)gh

      • Solving for VV:
        V=2ghV = \sqrt{2gh}

Practical Measurements

  • Measure the mass of both the waterball and the waterbottle.

  • Determine the swing distance (d) after the collision, using geometry to find h:

    • (L−h)2+d2=L2(L-h)^2 + d^2 = L^2

    • Simplifying with small angle assumptions:
      h=d22Lh = \frac{d^2}{2L}

Uncertainty Propagation

  • Consider standard deviation and the instrumental uncertainty of each measurement to report the mass of your waterball accurately.

  • Report: Mwaterball=m±uncertaintyM_{waterball} = m \pm uncertainty for mass with respective percent uncertainties.

Predictive Analysis

  • Use the initial velocity measurements to predict where the waterball will land if launched horizontally from height H using:

    • Range equation:
      D=vg×2HD = \frac{v}{g} \times \sqrt{2H}

    • Include uncertainty in predictions.

Experiment Execution

  • Launch waterballs multiple times to check for consistency.

  • Analyze results based on predicted landing positions versus actual results.

  • Explore the balance between uncertainty in measurements and predicted outcomes.

  1. Momentum conservation equation: mv=(m+M)Vmv = (m + M)V

  2. Solve for projectile velocity: v=(m+M)Vmv = \frac{(m + M)V}{m}

  3. Kinetic Energy and Potential Energy relationship: KE=PEKE = PE

  4. Equating energies: 12(m+M)V2=(m+M)gh\frac{1}{2}(m + M)V^2 = (m + M)gh

  5. Solving for V: V=2ghV = \sqrt{2gh}