Collision Simulations

Objectives

One-dimensional collisions are an excellent topic for study in an introductory physics course. The analysis is straightforward (but not necessarily simple!) and simple equipment permits an experimental test of important principles like conservation of energy and momentum.

In this lab, you and your group will explore collisions in one and two dimensions. The lab consists of two parts.

  1. Observe 1D collisions. Make predictions and qualitative observations of collisions in one dimension.
  2. Analyze 2D collisions. Analyze conservation of energy and momentum in two dimensions.

Record your setup, procedure, data, and observations in your lab notes. Keep detailed notes. Data for this lab will include tables and plots. In your memo, summarize your findings and describe any conclusions you draw from them.

Observe 1D Collisions

In this first portion of the lab, we will focus on qualitative observations. Open the Collisions lab from Andrew Duffy. Use it to investigate the following collisions. For each, follow the three-step process:

  • Predict. What do you expect to happen?
  • Observe. What actually happened? Does the same thing happen every time?
  • Analyze. Resolve any conflicts between your predictions and observations.

Discuss any surprises or easy predictions in your memo. Try to draw some general conclusions about the patterns you observe, too.

Elastic Collisions

For the following, set “Elasticity of the collision” to 1.0. The carts will collide and bounce off of each other.

  1. Equal Masses, One Stationary. One cart moves to the right. The other is at rest.
  2. Equal Masses, Head On. One cart moves to the right. The other moves to the left.
  3. Equal Masses, Same Direction. One cart moves quickly. The other moves slowly.
  4. Different Masses, One Stationary. A large cart moves to the right. A small cart is at rest.
  5. Different Masses, One Stationary, Again. A small cart moves to the right. A large cart is at rest.
  6. Different Masses, Head On. A large cart moves to the right. A small cart moves to the left.
  7. Different Masses, Same Direction. A large cart moves quickly. A small cart moves slowly.
  8. Different Masses, Same Direction, Again. A small cart moves quickly. A large cart moves slowly.

Inelastic Collisions

For the following, set “Elasticity of the collision” to 0.0. The carts will collide and stick together.

  1. Equal Masses, Head On, Same Speed. One cart moves right. The other moves left at the same speed.
  2. Different Masses, Head On. A large cart moves right. A small cart moves left.
  3. Different Masses, One Stationary. A large cart moves right. A small cart is at rest.
  4. Different Masses, One Stationary, Again. A small cart moves right. A large cart is at rest.
  5. Different Masses, Same Direction. A large cart moves quickly. A small cart moves slowly.
  6. Different Masses, Same Direction, Again. A small cart moves quickly. A large cart moves slowly.

Analyze 2D Collisions

Conservation of Momentum — In any collision, Newton’s Second and Third Laws imply that momentum should be conserved. The momentum of a single particle is \(\vec{p} = m\vec{v}.\) If momentum is conserved, then we expect the total momentum to be constant in a collision. In particular, the momentum before and after a collision between two objects (\(A\) and \(B\)) should be the same: \[\vec{P}_{\tt total} = M_{A} \vec{V}_{A1} + M_{B} \vec{V}_{B1} = M_{A} \vec{V}_{A2} + M_{B} \vec{V}_{B2}\] Momentum is a vector. Conservation of momentum implies that, in two dimensions, the horizontal and vertical components of the total momentum are conserved independently.

Kinetic Energy — The kinetic energy of a particle is \[E = \dfrac{1}{2} MV^2 = \dfrac{P^2}{2M}\] In a perfect elastic collision, total kinetic energy is conserved. In an inelastic collision, a fraction of kinetic energy is converted to other forms — often heat. We can characterize a collision by looking at the change in energy: \[ E_1 = \dfrac{1}{2} M_A V_{A1}{}^2 + \dfrac{1}{2} M_B V_{B1}{}^2 \qquad \qquad \qquad E_2 = \dfrac{1}{2} M_A V_{A2}{}^2 + \dfrac{1}{2} M_B V_{B2}{}^2 \qquad \qquad \qquad \Delta E = E_2 - E_1 \]

Collect Data

In this portion of the lab, the goal is to determine the degree to which momentum and kinetic energy are conserved in a simulated collision of curling stones. Open the Curling Stone Momentum Lab from The Physics Aviary. Play with the simulation to figure out how it works. When you are ready, begin collecting data.

Take data for 3 different collisions. For the first, use the following parameters in the table. You can choose the parameters for the other two as long as (1) the masses are not equal, and (2) the separation is greater than \(0\,\mathsf{cm}\).

Trial Separation [\(\mathsf{cm}\)] Red Mass [\(\mathsf{kg}\)] Yellow Mass [\(\mathsf{kg}\)] Red Speed [\(\mathsf{m/s}\)]
1 50 16 16 4
2
3

Record the final speed and angle of both stones, and carry out the following analysis for each collision.

Analysis: Conservation of Momentum

Use trigonometry to compute the initial and final momentum of each stone in the horizontal direction and the vertical direction. Calculate the change in momentum. The following tables may be a useful way to organize the calculations.

Trial \(P_{\tt red, in, x}\) \(P_{\tt yellow, in, x}\) \(P_{\tt total, in, x}\) \(P_{\tt red, out, x}\) \(P_{\tt yellow, out, x}\) \(P_{\tt total, out, x}\) \(\Delta P_{\tt total, x}\)
1
2
3
Trial \(P_{\tt red, in, y}\) \(P_{\tt yellow, in, y}\) \(P_{\tt total, in, y}\) \(P_{\tt red, out, y}\) \(P_{\tt yellow, out, y}\) \(P_{\tt total, out, y}\) \(\Delta P_{\tt total, y}\)
1
2
3

Note: Direction matters! Make sure you account for the direction of the outgoing stones when calculating components of momentum. If one stone is moving upward and the other is moving downward, their vertical components of momentum should have opposite signs. If one stone is moving right and the other is moving left, their horizontal components of momentum should have opposite signs.

Analysis: Conservation of Energy

Compute the initial and final kinetic energy of each stone. Calculate the change in kinetic energy. The following table may be a useful way to organize the calculations.

Trial \(E_{\tt red, in}\) \(E_{\tt yellow, in}\) \(E_{\tt total, in}\) \(E_{\tt red, out}\) \(E_{\tt yellow, out}\) \(E_{\tt total, out}\) \(\Delta E_{\tt total}\)
1
2
3
  • Based on your calculations, is momentum conserved in this simulation?
  • Based on your calculations, is this a simulation of an elastic or an inelastic collision?

Further Exploration

The following activities are not required. If you have the time and the inclination, you may carry out these investigations for extra credit. You are also free to use any of the materials available in the lab to carry out your own investigations. You need to keep a record of your work (setup, procedure, data, analysis) to get credit for your efforts.

Ballistic Pendulum. The total momentum and energy in a system are conserved, but energy can change forms. In a ballistic pendulum, the initial kinetic energy of a bullet is partially converted to gravitational potential energy. The rest is dissipated as heat. Open the Ballistic Pendulum simulation from oPhysics and investigate this system.

Measure the change in height as the pendulum swings. Use the mass of the block and bullet to determine its total energy after the collision, and compute the fraction of initial kinetic energy lost in the collision. Is this an elastic or an inelastic collision?