# Momentum and energy equations for a car rolling over a step

In summary, the conversation discusses a model car rolling down a track and encountering a step. The problem at hand is figuring out how much of the translational and rotational energy is converted into a vertical "bounce" when different wheels hit the step. The conversation also touches on the factors that may affect the energy loss, such as the placement of the center of mass and the length of the wheelbase. The solution may require calculating the moment of inertia and using conservation of momentum and angular momentum.
A model car with a main body mass M and four wheels each with mass m is rolling down a track at velocity v, when it encounters a step of height b, which is less than the wheel radius r. The wheels are rolling, not sliding, and we know the moment of inertia around their respective centers and thus the angular velocity and angular momentum of each. The distance & direction of each pair of axles from the COM is known. (See schematic)

Presuming frictionless axles and a totally elastic collision, I am trying to use conservation of momentum and angular momentum to figure out how much of the translational and rotational energy is converted into a vertical "bounce" when the front pair of wheels hits the step, and later when the rear wheels hit the step. The COM is slightly in front of the rear axles, and we can presume that the tipping of the car will not cause the rear of the main body to drop far enough to hit the step, nor will the first impact have enough energy to flip the front up over the back.

I have only seen problems like this for a single wheel, nothing for a car that has a COM far behind the wheel center. I'm confused about how to write the equations when there are multiple masses and multiple (car / wheel) types of rotation. Do I have to calculate all angular momentum about a single point? If so, is the impact point the best choice?

I think this can be done with instantaneous impulses and the conservation of momentum, but I'm having a tough time drawing the diagram and the forces, velocities and angular velocities before and after each impact. Can someone help me set up the system of equations to solve?

P.S. we can assume a flat section of track, if it makes any difference, with the acceleration of gravity pointing straight downward. This may affect how long it takes for the bounce to dampen out, but b<<r, so I'm considering that the magnitude of the bounce will be small, almost imperceptible - with the car coming back to a normal roll very soon after the rear wheels get over the step.

#### Attachments

• car step.png
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I think the solution will require me to calculate the moment of inertia of the main body around some axis, as some of the angular momentum at the wheels gets converted to angular momentum of the main body as the front end bounces up.

That is further complicated by the fact that the main body is not a constant density. For simplicity, it can be approximated by two sections (with rectangular cross-section) as shown in this revised schematic. The rear section will have a higher density than the front, for reasons that aren't really related to this problem.

I am hoping to find (for a given mass and overall length) a relationship between wheelbase (L1+L2), COM placement, and the energy loss response to the step impulse. It appears that a long wheelbase and a low COM will result in the lowest energy loss, as that gives the minimum angle between the impact point and the COM, and thus the lowest vertical component to the impulse.

#### Attachments

• car step 2.png
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## 1. What is the difference between momentum and energy equations?

The momentum equation describes the motion of an object in terms of its mass and velocity, while the energy equation describes the energy changes that occur during the motion of an object.

## 2. How do I calculate the momentum of a car rolling over a step?

To calculate the momentum of a car rolling over a step, you will need to know the mass of the car and its velocity before and after the step. The momentum can be calculated using the formula p = m * v, where p is the momentum, m is the mass, and v is the velocity.

## 3. What is the conservation of momentum and energy?

The conservation of momentum states that in a closed system, the total momentum remains constant, meaning that the initial momentum of the system is equal to the final momentum. The conservation of energy states that energy cannot be created or destroyed, but can only be transferred or transformed from one form to another.

## 4. How does the height of the step affect the energy equation for a car rolling over it?

The height of the step will affect the potential energy of the car, which is one of the components of the total energy equation. The higher the step, the greater the potential energy of the car will be, resulting in a larger change in energy during the rolling motion.

## 5. How do friction and air resistance impact the momentum and energy equations for a car rolling over a step?

Friction and air resistance will decrease the momentum and energy of the car as it rolls over the step. This is because these forces act in the opposite direction of the car's motion, causing a decrease in its velocity and therefore its momentum. The energy equation will also be affected as some of the car's kinetic energy is converted into thermal energy due to friction and air resistance.

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