Collisions

Objectives

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

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

  1. Observe collisions. Make predictions and qualitative observations of collisions.
  2. Investigate collisions. Take data for both elastic and inelastic collisions.

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 Collisions

Later in this lab, we will try to make precise measurements of collisions. In this first portion of the lab, we will focus on qualitative observations. Using motion carts, 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, use magnets or springs so that 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, use velcro so that 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.

Study the Equipment

Take some time to familiarize yourself with the equipment. We will be using the Vernier motion carts. You can use a pair of motion sensors, or the carts with internal sensors. Keep a record of your investigation in your notes.

Set up a pair of motion detectors (one for each cart) or link a pair of Bluetooth carts to the Graphical Analysis software. Investigate the cart, track, and launching mechanism.

  • Can you plot the position of both carts on the same axes? Velocity?
  • Figure a method to securely add mass to the carts.

Once you are able to track both carts, investigate the following questions. Take data. Answer each question with a measurement or experimental observation.

  • Does a cart at rest remain at rest? Try to eliminate any drift.
  • How significant are friction and drag forces? I.e., how much does a cart slow down?
  • What percent of its speed does a cart lose over \(50\,\mathsf{cm}\)?

Collisions

In an elastic collision, two objects collide and rebound. In an inelastic collision, they collide and stick together.

Conservation of Momentum — In any collision, Newton’s Second and Third Laws imply that total 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 gliders (\(A\) and \(B\)) should be the same: \[\vec{P}_{\tt total} = M_{A} \vec{V}_{A1} + M_{B} \vec{V}_{B1} = M_{A} \vec{V}_{A2} + M_{B} \vec{V}_{B2}\] Note that momentum is a vector, so direction is important.

Kinetic Energy — The kinetic energy of a particle is \(E = \frac{1}{2} mv^2\). In a perfect elastic collision, the total kinetic energy is conserved. In an inelastic collision, some kinetic energy is converted to other forms — most often, heat.

In this final portion of the lab, the goal is to experimentally determine the degree to which momentum and kinetic energy are conserved. You should collect data for two elastic collisions and two inelastic collisions. Record the mass of the gliders or carts involved in each collision, and take determine the initial and final velocities of each cart or glider.

  1. Elastic Collisions — Use magnets or springs to study elastic collisions. Collect data for at least one collision with equal masses and one collision for unequal masses.

  2. Inelastic Collisions — Use velcro to study elastic collisions. Collect data for at least one collision with equal masses and one collision for unequal masses.

  • Do your data support the claim that total momentum is conserved?
  • Are any of the collisions you observed elastic?
  • What fraction of kinetic energy is lost in each collision?

Organizing Data

Organizing data for this portion of the lab can be challenging. The following tables may be useful guides.

The raw data for this experiment consists of masses and velocities. The following table will contain all of the data you need. Be sure to record the initial and final directions of each cart, not just the speed. This will determine the sign of the momentum. A sketch of “Before” and “After” is a great way to record this information.

Trial \(M_A\) [\(\mathsf{kg}\)] \(M_B\) [\(\mathsf{kg}\)] \(V_{A1}\) [\(\mathsf{m/s}\)] \(V_{A2}\) [\(\mathsf{m/s}\)] \(V_{B1}\) [\(\mathsf{m/s}\)] \(V_{B2}\) [\(\mathsf{m/s}\)]
Elastic 1
Elastic 2
Inelastic 1
Inelastic 2
Variable Meaning
\(M_A\) mass of Cart A
\(M_B\) mass of Cart B
\(V_{A1}\) initial velocity of Cart A
\(V_{A2}\) initial velocity of Cart B
\(V_{B1}\) final velocity of Cart A
\(V_{B2}\) final velocity of Cart B

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.

Elasticity: A common measure of elasticity in collisions is the coefficient of restitution or COR: \[e = \left|\dfrac{V_{A2} - V_{B2}}{V_{A1} - V_{B1}}\right|\] where \(V_{A1}\) and \(V_{B1}\) are the incoming velocities and \(V_{A2}\) and \(V_{B2}\) are the outgoing velocities of two objects of equal mass. \(e=1\) for a perfectly elastic collision and \(e=0\) for a perfectly inelastic collision.

Compute the coefficient of restitution for each of the collisions you studied in Part 3. How does your elastic collision compare with the theoretical limit of \(e=1\)?

Friction: Accurately measure the friction and drag forces by studying the motion of carts as they move along the track. Study the dependence of these forces on speed. Try to determine the nature of these forces using your data.

I.e., do you see a constant force? A force that grows linearly with speed? A force that grows quadratically with speed? Something else?

Force Force Law
friction \(f = -\mu N\)
viscous drag \(f = -bv\)
air resistance \(f = -cv^2\)