A collision experiment is a physics experiment in which moving carts or objects are collided and their velocities before and after the collision are measured to investigate conservation of momentum and changes in kinetic energy.
In a system where the effects of external forces can be neglected, the total momentum of the system is conserved before and after a collision. Whether kinetic energy is conserved depends on the type of collision. It is approximately conserved in an elastic collision, whereas in an inelastic collision some of it is converted into deformation, sound, heat, and other forms of energy.
In an experimental report, it is important not only to compare momentum before and after the collision, but also to discuss the coefficient of restitution, changes in kinetic energy, rail friction, misalignment of the collision direction, velocity measurement, and how these factors affected the results.
This article considers a one-dimensional collision experiment involving two carts on a horizontal track and summarizes the items to record in the results, sample experimental values, calculation methods, sources of error, example result statements, and discussion examples that can be used in a report.
Note:
This article is a study reference for discussing results obtained in physics experiments at universities and similar institutions. The experimental values shown are examples for explanation and are not actual measurement results.
For handling carts, tracks, springs, magnets, adhesive materials, photogates, and all safety precautions, always follow the laboratory manual and the instructions of your instructor or teaching assistant.
What Is a Collision Experiment?
A collision is a phenomenon in which two or more objects exert large forces on one another over a short period of time, changing their velocities or directions of motion. Because the forces between the objects during the collision form an action-reaction pair, the total momentum of the system is conserved if the impulse due to external forces is sufficiently small.
Let the mass of object 1 be m1, its velocity before the collision be u1, and its velocity after the collision be v1, with corresponding quantities for object 2. Conservation of momentum is expressed as follows.
m1u1 + m2u2 = m1v1 + m2v2
Because velocity has direction, define a positive direction and include signs in the calculations. A cart moving opposite to the positive direction is assigned a negative velocity.
Elastic Collision
In an elastic collision, not only the total momentum of the system but also its translational kinetic energy is conserved.
1/2 m1u12 + 1/2 m2u22 = 1/2 m1v12 + 1/2 m2v22
Inelastic Collision
In an inelastic collision, momentum is conserved but translational kinetic energy is not. A collision in which the two objects stick together and move as one after impact is called a perfectly inelastic collision.
m1u1 + m2u2 = (m1 + m2)v
Coefficient of Restitution
The ratio of relative speeds after and before a collision is called the coefficient of restitution e. For a one-dimensional collision, it is expressed as follows.
e = (v2 − v1) / (u1 − u2)
For an ideal elastic collision, e = 1, while for a perfectly inelastic collision, e = 0.
Main Items to Record in the Results
- Masses of cart 1 and cart 2
- Positive direction defined along the track
- Velocity of each cart before the collision
- Velocity of each cart after the collision
- Type of collision
- Total momentum before the collision
- Total momentum after the collision
- Difference in momentum and momentum retention percentage
- Total kinetic energy before the collision
- Total kinetic energy after the collision
- Kinetic-energy retention percentage or decrease percentage
- Coefficient of restitution
- Number of measurements and average values
- Presence of sound, deformation, vibration, or rotation during the collision
If a cart moves opposite to the positive direction, record not only the magnitude of its velocity but also its sign. The sign is important when calculating momentum, whereas kinetic energy is always positive because velocity is squared.
Sample Experimental Conditions
| Item | Sample Condition |
|---|---|
| Apparatus | Two low-friction carts on a horizontal track |
| Mass of cart 1 | 0.500 kg |
| Mass of cart 2 | 0.500 kg |
| Positive direction | Direction from cart 1 toward cart 2 |
| Velocity measurement | Photogate or video analysis |
| Elastic collision | Spring or magnetic bumpers |
| Perfectly inelastic collision | Carts connected after impact using hook-and-loop fasteners or similar material |
| Number of measurements | 3 for each condition |
Sample Experimental Values
Elastic Collision Between Equal-Mass Carts
The following are sample values for a moving cart 1 colliding with a stationary cart 2 of equal mass.
| Trial | u1 (m/s) | u2 (m/s) | v1 (m/s) | v2 (m/s) | Momentum Before (kg·m/s) | Momentum After (kg·m/s) |
|---|---|---|---|---|---|---|
| 1 | 0.802 | 0.000 | 0.031 | 0.765 | 0.401 | 0.398 |
| 2 | 0.798 | 0.000 | 0.028 | 0.763 | 0.399 | 0.396 |
| 3 | 0.805 | 0.000 | 0.034 | 0.768 | 0.403 | 0.401 |
When equal-mass carts undergo an ideal elastic collision, the initially moving cart nearly stops and the initially stationary cart begins moving at approximately the original speed of the first cart.
Perfectly Inelastic Collision Between Equal-Mass Carts
| Trial | u1 (m/s) | u2 (m/s) | Common Velocity After Collision v (m/s) | Momentum Before (kg·m/s) | Momentum After (kg·m/s) |
|---|---|---|---|---|---|
| 1 | 0.804 | 0.000 | 0.397 | 0.402 | 0.397 |
| 2 | 0.799 | 0.000 | 0.395 | 0.400 | 0.395 |
| 3 | 0.807 | 0.000 | 0.399 | 0.404 | 0.399 |
When equal-mass carts undergo a perfectly inelastic collision and stick together, the theoretical common velocity is approximately half the initial velocity of cart 1.
Calculation Methods
Momentum Before and After the Collision
For the first elastic-collision trial, the total momentum before the collision is calculated as follows.
pbefore = m1u1 + m2u2
= 0.500 × 0.802 + 0.500 × 0.000
= 0.401 kg·m/s
The total momentum after the collision is:
pafter = m1v1 + m2v2
= 0.500 × 0.031 + 0.500 × 0.765
= 0.398 kg·m/s
Momentum Retention Percentage
Momentum retention (%) = pafter / pbefore × 100
= 0.398 / 0.401 × 100
= approximately 99.3%
Because momentum includes direction, signed velocity values must be used when motion in the opposite direction is involved.
Kinetic Energy in the Elastic Collision
For the first elastic-collision trial, the total kinetic energy before the collision is:
Kbefore = 1/2 m1u12 + 1/2 m2u22
= 1/2 × 0.500 × 0.8022
= 0.161 J
The total kinetic energy after the collision is:
Kafter = 1/2 × 0.500 × 0.0312 + 1/2 × 0.500 × 0.7652
= approximately 0.147 J
Kinetic-Energy Retention Percentage
Kinetic-energy retention (%) = Kafter / Kbefore × 100
= 0.147 / 0.161 × 100
= approximately 91.3%
In a real elastic-collision experiment, translational kinetic energy may not be perfectly conserved because of friction, sound, vibration, deformation of the bumpers, and other effects.
Common Velocity in a Perfectly Inelastic Collision
For the first perfectly inelastic collision trial, the theoretical common velocity follows from conservation of momentum:
v = (m1u1 + m2u2) / (m1 + m2)
= (0.500 × 0.804 + 0.500 × 0.000) / (0.500 + 0.500)
= 0.402 m/s
The experimental value of 0.397 m/s is close to the theoretical value of 0.402 m/s.
Kinetic Energy in a Perfectly Inelastic Collision
Kbefore = 1/2 × 0.500 × 0.8042
= approximately 0.162 J
Kafter = 1/2 × (0.500 + 0.500) × 0.3972
= approximately 0.0788 J
Kinetic-energy retention = 0.0788 / 0.162 × 100
= approximately 48.6%
For a perfectly inelastic collision between equal masses when one object is initially at rest, even under ideal conditions the translational kinetic energy after the collision is approximately half of that before the collision.
Coefficient of Restitution
For the first elastic-collision trial, the coefficient of restitution is:
e = (v2 − v1) / (u1 − u2)
= (0.765 − 0.031) / (0.802 − 0.000)
= approximately 0.915
Because the coefficient of restitution is less than 1, this collision was not perfectly elastic and involved a small energy loss.
Main Sources of Error
| Source of Error | Effect on Measurement | Likely Trend in Results | Improvement or Check |
|---|---|---|---|
| Friction between track and wheels | Reduces cart velocity before and after collision | Total momentum after collision tends to appear smaller | Level the track and measure velocity immediately before and after collision |
| Track inclination | Gravity component accelerates or decelerates carts | Momentum changes systematically with time | Check that the track is level |
| Misaligned collision direction | Carts rotate and motion is no longer one-dimensional | Translational momentum and energy appear smaller | Align the centers of the bumpers |
| Velocity measurement error | Affects calculations of momentum and kinetic energy | Error is especially amplified in energy because velocity is squared | Place photogates near the collision point and repeat measurements |
| Cart mass measurement error | Affects both momentum and kinetic energy | Calculated values shift systematically | Measure mass including all attachments |
| Deformation during collision | Converts part of kinetic energy into internal energy | Kinetic energy after collision decreases | Use highly elastic bumpers |
| Sound and vibration | Converts some translational kinetic energy into other forms | Energy retention decreases | Keep collision conditions consistent |
| Cart rotation | Transfers some translational kinetic energy into rotational motion | Energy calculated from translational velocity appears smaller | Make carts collide head-on |
| Measurement positions too far from collision point | Friction acts while carts travel through the measurement interval | Deviation from conservation laws increases | Measure velocities close to the collision point |
Why Momentum Does Not Match Perfectly
Conservation of momentum holds when the impulse of external forces acting on the system is negligible. In an actual experiment, friction between the track and wheels, air resistance, and a slight track inclination act as external forces.
In addition, velocity is measured slightly before or after the instant of collision rather than exactly at that instant, so friction also acts while the carts travel to the measurement points. Consequently, the total momentum after the collision may be slightly smaller than before it.
Why Kinetic Energy Decreases
Momentum is a vector quantity with direction, and internal interactions alone do not change the total momentum of the system. Translational kinetic energy, however, can be converted into deformation, sound, heat, vibration, rotational motion, and other forms during a collision.
Therefore, even when momentum is approximately conserved, translational kinetic energy is not necessarily conserved. In a perfectly inelastic collision in particular, the objects move together after impact, so the decrease in kinetic energy is large.
Why the Sign of Velocity Matters
Because momentum includes the direction of velocity, the velocity of an object moving opposite to the positive direction must be treated as negative. Using only speed magnitudes would prevent the momenta of oppositely moving objects from being added correctly.
For kinetic energy, on the other hand, velocity is squared, so the value is positive regardless of whether the velocity is positive or negative.
Example of How to Write the Results
Using two carts of mass 0.500 kg, velocities before and after collision were measured for an elastic collision and a perfectly inelastic collision.
In the first elastic-collision trial, the total momentum was 0.401 kg·m/s before the collision and 0.398 kg·m/s after it, giving a momentum retention of approximately 99.3%. The kinetic energy was 0.161 J before and 0.147 J after the collision, giving a kinetic-energy retention of approximately 91.3%. The coefficient of restitution was approximately 0.915.
In the first perfectly inelastic collision trial, total momentum was 0.402 kg·m/s before the collision and 0.397 kg·m/s after it. Kinetic energy decreased from approximately 0.162 J before the collision to approximately 0.0788 J after it, giving a retention of approximately 48.6%.
Points to Connect to the Discussion
- How closely did total momentum agree before and after the collision?
- What was the momentum retention percentage?
- How well was kinetic energy conserved in the elastic collision?
- Into what forms was the lost translational kinetic energy converted in the perfectly inelastic collision?
- Was the coefficient of restitution consistent with the type of collision?
- Were positive and negative velocity signs handled correctly?
- Did track friction or inclination affect the results?
- Was the collision one-dimensional without cart rotation?
- Were velocities measured sufficiently close to the collision point?
- Were cart masses measured including attachments?
Discussion Example
In this experiment, two carts were collided to investigate changes in momentum and kinetic energy in elastic and perfectly inelastic collisions.
In the first elastic-collision trial, total momentum was 0.401 kg·m/s before the collision and 0.398 kg·m/s after it, giving a momentum retention of approximately 99.3%. Because these values agree closely, conservation of momentum can be considered to hold within the experimental accuracy.
In contrast, kinetic energy was 0.161 J before and 0.147 J after the collision, corresponding to a retention of approximately 91.3%. Although kinetic energy is conserved in an ideal elastic collision, it was not perfectly conserved here. This is likely because part of the translational energy was converted into other forms by collision sound, bumper deformation, vibration of the carts and track, and friction between the wheels and track.
The coefficient of restitution was approximately 0.915, which is less than the ideal elastic-collision value of 1. This also indicates that the collision was not perfectly elastic and involved a small energy loss.
For the perfectly inelastic collision, momentum before and after the collision was 0.402 kg·m/s and 0.397 kg·m/s, respectively, so momentum was approximately conserved. Kinetic energy, however, decreased from approximately 0.162 J to approximately 0.0788 J, giving a retention of approximately 48.6%.
When one of two equal-mass carts is initially stationary and the carts stick together after collision, conservation of momentum predicts a common velocity approximately half the initial velocity. Even ideally, the translational kinetic energy after such a collision is half the initial value. The lost translational kinetic energy is converted into internal energy such as deformation of the carts, vibration of the coupling, sound, and heat.
Possible reasons why momentum before and after the collision did not match perfectly include friction between the track and wheels, a slight inclination of the track, velocity measurement error, and misalignment of the collision direction. In particular, an off-center collision can cause cart rotation, meaning that one-dimensional velocity alone no longer describes all of the motion.
To improve measurement accuracy, the track should be carefully leveled and velocities measured immediately before and after the collision point. The bumper centers should also be aligned to prevent cart rotation, and repeated measurements should be averaged.
Discussion Example When Momentum After the Collision Was Smaller
If the total momentum after the collision was smaller than before it, one possible cause is the impulse of external forces such as friction between the track and wheels or air resistance. If the post-collision velocity was measured at a point away from the collision location, the cart may also have slowed before reaching the measurement point, causing the momentum immediately after the collision to be underestimated.
Discussion Example When Kinetic Energy Decreased Substantially in an Elastic-Collision Trial
If kinetic energy decreased substantially despite the collision being intended as elastic, possible causes include deformation of the bumpers, collision sound, vibration of the carts or track, and cart rotation. In particular, if the collision is off-center, some translational kinetic energy is transferred to rotational motion, so kinetic energy calculated only from the linear velocity appears smaller.
Summary
In a collision experiment, the masses of the carts and their velocities before and after collision are measured to determine whether the total momentum of the system is conserved.
In an ideal elastic collision, both momentum and kinetic energy are conserved. In an inelastic collision, momentum can remain conserved while translational kinetic energy is converted into deformation, sound, heat, vibration, and other forms.
In the discussion, it is important to relate momentum retention, kinetic-energy retention, the coefficient of restitution, velocity signs, friction, track inclination, collision alignment, cart rotation, and measurement positions to the experimental results.
