# Evaluating Crumple Zone Performance

This lab is designed to align with AAOT science outcome #1: Gather, comprehend, and communicate scientific and technical information in order to explore ideas, models, and solutions and generate further questions.

## Materials:

• writing utensil
• calculator
• digital device with spreadsheet program
• digital device with internet access

## ObjectiveS

1. Theoretically investigate the effectiveness of crumple zones by applying the Law of Conservation of Energy to model elastic and inelastic collisions.
2. Apply the model to predict the relative magnitude of average and peak forces during collisions with and without crumple zones.
3. Experimentally investigate the effectiveness of crumple zones by measuring average and peak forces with and without crumple zones.
4. Compare the model predictions with the experimental results.
5. Apply the Impulse-Momentum Theorem to determine change in velocity from force vs. time data.

## Collision Modeling

For the simplified case of a car hitting a solid barrier that doesn’t move, let’s compare the force peak applied to occupants for the theoretical case of a perfectly elastic car and a perfectly designed crumple zone. We will simplify our problem by analyzing the case of a car hitting a solid immovable barrier. We start by analyzing the first half of the collision (as the car/person comes to rest before bouncing back) and including the car and barrier in our system. We will assume the crumple zone provides a constant resistive force as it crumples and dissipates kinetic energy into thermal energy.

### Crumple Zone Model

$0 = \Delta PE +\Delta KE$ +\Delta TE

Internal work is done on the car-barrier system to convert KE to TE and we are assuming that work was done by a constant force over the total crumple distance, $x$. In that case $\Delta TE = Fx$. No energy is stored as PE:

$0 = 0 +\Delta KE + Fx$

Expanding the change in kinetic energy and setting the final kinetic energy to zero:

$0 = 0-\frac{1}{2} mv_i^2 +Fx$

1) For this ideal case of the constant force, the average and peak force exerted are the same. Symbolically solve for that force. Show your work.

### Elastic Model

$0 = +\Delta KE$ +\Delta TE

In the elastic collision internal work is done on the car-barrier system to convert KE to elastic PE. No energy is stored as TE. Expanding the change in kinetic energy and setting the final kinetic energy to zero:

$0 = \Delta PE + 0-\frac{1}{2} mv_i^2$

We will model the elastic material as a spring, so change in elastic PE is $\Delta PE =(1/2)kx^2$.

$0 = \frac{1/2}kx^2 -\frac{1}{2} mv_i^2$

Expanding the $x^2$ allows us to introduce the Force into the equation.

$0 = \frac{1/2}kxx +\frac{1}{2} mv_i^2$

We recognize that for a spring $F = kx$ and that $x$ and make that substitution:

$0 = \frac{1/2}Fx +\frac{1}{2} mv_i^2$

2) The force is applied at maximum compression distance $x$ is the peak force. Symbolically solve for the peak force. Show your work.

3) Compare the peak force of the elastic collision and the peak force of the dissipative collision. Which is larger, and by how much?

4) The spring force increases linearly with compression distance, so the average force exerted by a spring will be half of the maximum force exerted at maximum compression. Therefore the average force in the elastic collision is half of what you found for the peak force.  How does the average force compare between the elastic and dissipative collisions?

5) Does what you found above agree with the Work-Energy Theorem? Explain. [Hint: The first half of both collisions caused the car to experience the same change in kinetic energy over the same distance.]

Assuming the seatbelt holds the person firmly to the seat so that they always have the same velocity as the car and come to a stop over the same distance (and time). Therefore our model of the forces applies to both the car and the person. To find the forces on the person we would simply use their mass in the equations instead of the car mass.

6) A car with a ridged frame that does not permanently deform will act similarly to a spring, except that the compression happens over a very short distance. Considering that behavior, rank the size of the peak collision force for elastic cars, rigid frame cars, and cars with crumple zones as predicted by our model. Explain your reasoning.

7) Does our model predict that crumple zones are effective at reducing collision forces? Explain.

Real collisions can be more complex than our idealized model. For example, the colliding objects may have similar mass, the case of the elastic collision will not be perfectly elastic, and the crumple zone will likely not produce perfectly constant force. Friction from the bottom of the seat also acts on the occupant. We are assuming the model still captured the overall behavior of each collision type to provide us with the correct ranking of peak forces and the correct conclusion about the effectiveness of crumple zones,  but we still want experimental verification.

## Experimental collision Force Testing

### Data Collection

Collision experiments were performed for a rigid frame, a springy frame, an attempted crumple zone built from soda can aluminum, and finally a crumple zone built from aluminum foil.  The velocity of the cart was measured by the cart’s optical sensor and the force on the simulated occupant of the cart was measured with a force sensor. All of the necessary data is available in an online spreadsheet. You can copy and paste the data into your own sheet for analysis.

### Data Analysis

8) Create a single graph containing plots of the the velocity vs. time data for all 3 trials shown in the datasheet. This will allow you to quality check your data. Make sure you have a complete dataset that includes the maximum speed reached by the cart for each trial. Be sure to name your graph and label the axes, including units. Also be sure to add a legend so you or anyone else can identify which dataset in your plot is which trial.

9) Create a single graph containing plots of the the force vs. time data for all 3 trials shown in the datasheet. This will allow you to quality check your data. Make sure you have a complete dataset that includes the maximum speed reached by the cart for each trial. Be sure to name your graph and add axis labels with correct units. Also be sure to add a legend so you or anyone else can identify which dataset in your plot is which trial.

10) Use the MAX spreadsheet function to find the peak speed for each trial. (The datasheet contains empty rows at the top for you to apply this function). Are the collision speeds for each trial all the same to within a few percent? Show your calculations below.

11) Based on the experimental data, rank the collisions by peak force applied to the  occupant, from lowest to highest.

12) Change the scale on the force graph to zoom in on the impulses for the two non-rigid collisions.  Describe the shape of the impulses for the spring collision and crumple collision.

13) Did the crumple zone act as designed and provide a (nearly) constant force during the collision?

#### Max and Average Force Hypotheses

14) Use the MAX and AVERAGE spreadsheet functions to find the peak force and average force for each trial. (The datasheet contains empty rows at the top for you to apply this function). In the space below, record the peak force and average force for each type of collision.

## conclusions

15) Considering your results, when comparing rigid frames, spring systems, or crumple zones, which is the best safety system?

## Further Questions

16) What changes could we make to our experimental methods to reduce uncertainty in our measurements? Provide a detailed explanation. If you suggest new equipment, be sure to including a description of how the equipment would work.

17) What changes could we have made to our experiment to more accurately recreate a real collision between cars?

Overall, the crumple zone decreases the impact speed between your skull and the airbag while the airbag decreases the impact speed between your brain and skull.

During our experiment we use the motion sensor to measure the bounce-back speed of the cart because the fan was in the way. However, we can use the Impulse-Momentum Theorem to find the change in velocity for the occupant of the vehicle. The larger the change in velocity, the more bouncy the collision was. The cart mass doesn’t change during the collision, so the Impulse Momentum Theorem looks like this: $\bold{F_{ave}}\Delta t = m\bold{\Delta v}$.

18) Solve the impulse momentum theorem for the change in velocity.

19) Find the impulse on the car for each collision by multiplying the average force for each collision by the duration of the collision.(You will notice in the data that all of the collisions have the same duration. Record the impulse values here:

20) The mass of the simulated occupant was 0.05 kg. Calculate the change in velocity for each collision. Show your work.

21) Rank the collisions by the magnitude of change in velocity.

22) Does the result of analyzing the change in velocity add support to our conclusion about the effectiveness of crumple zones, or not? Explain your reasoning.