Modelling a Collision

Physics is an interesting science. It is the most rigorous of the sciences, and as such, most every observation, rule, or law in physics can be put in terms of math. We can then use this math to predict what would happen under specific circumstances.

Take the collision of two objects, for example. There are two laws in particular that we can use to predict the outcome of the collision. The first is the conservation of momentum, which tells us that, when there are no other outside forces acting on objects, the total momentum of all objects in a system will remain constant. In a similar vein, the conservation of energy tells us that, absent any input or use of energy from the outside, the total energy in a system will remain constant. We can use these two laws to predict the aftermath of a collision, provided we make a few stipulations.

Now, before we begin with the prediction, it should be pointed out that momentum and energy are different types of quantities. Momentum is a vector while energy is a scalar. A scalar is a quantity that measures one thing and one thing only. We are most familiar with scalar measurements. If we measure the length of something, the length we measure is a scalar: the objects has one length and one length only. If we measure an object’s weight, which we typically use as a proxy for the object’s mass, that is also a scalar: the weight of the object does not change regardless of where the object is or where it is facing.^[1] A vector, however, has both a quantity and a direction. Both the quantity and the direction together make up a vector. A good example is the difference between speed and velocity. Speed is a scalar and velocity is a vector, yet they both describe the same thing: how fast an object is moving. With speed, all we care about is the rate at which distance changes. For example, a car can drive 60 miles per hour (the amount of distance that changes per hour) in any direction and its speed will still be 60 miles per hour. A car driving North at 60 miles per hour has a speed of 60 miles per hour, a car driving East at 60 miles per hour has a speed of 60 miles per hour, a car driving South at 60 miles per hour has a speed of 60 miles per hour, and so forth. Velocity, however, contains both an object’s speed and its direction. If we say that a car driving North has a velocity of 60 miles per hour, then a car driving 60 miles per hour South has a velocity of -60 miles per hour. Why is the quantity negative? Because North is the chosen direction of velocity: going away from North (South) is going the opposite direction, so the velocity is negative. Note that North was arbitrarily chosen as the direction for velocity: we can make velocity be any direction that we like. But once we decide which is the direction of velocity, all other velocities have to be given relative to that direction.

Now, back to momentum and energy. Since momentum is a vector, it not only tells the magnitude of something, it also tells that object’s direction. Energy is a scalar, so for it, we are only interested in quantity. Now, momentum and energy are measured differently. Momentum is the velocity of an object times its mass. If we let 𝑚 represent the mass of an object and 𝑣 represent the velocity of an object, then its momentum is 𝑚𝑣. To calculate the energy of a moving object, we also need to know its mass and its velocity, but now the equation for energy is ½𝑚𝑣². Notice that, while a velocity, and therefore momentum, can be negative, since the velocity is squared, energy will always be positive.

Now, let us start to define how a collision will take place. Let us suppose that only two objects collide with one another. Let us suppose that the objects collide head-on. That is, the objects are moving directly toward each other: they do not collide at an angle. Stipulating that the objects collide head-on assures us that the objects will react by bouncing directly back from one another, rather than bouncing off at odd angles. This will make out calculations easier. We will allow ourselves to specify the mass of the two objects, which we will call 𝑚₁ and 𝑚₂, and the initial velocities of the two objects, which we will call 𝑣_1i and 𝑣_2i. The final velocities will be 𝑣_1f and 𝑣_2f, and these will be calculated. Finally, we will specify that the collision is perfectly elastic.

An elastic collision is one where the objects return to their original shape after the collision. Think of a rubber band. The rubber band is elastic: you can stretch it, but after letting it go, it returns back to its original shape. During a collision, the objects can be temporarily misshaped, but if the collision is elastic, the objects will return to their original shapes. Specifying that our collision is elastic is important because it means that no energy will be lost during the deformation of the objects. Say two clay balls were to collide with one another. Clay deforms very easily but it is not elastic. When they collide, both balls will deform but they will remain deformed. It takes energy to deform the balls, but since they do not rebound to their original shapes, that energy is not recovered. In other words, in a non-elastic collision, some energy is “lost” during the collision and never recovered.^[2] In an elastic collision, energy is used to deform the objects, but since they objects rebound, the energy is recovered as they snap back to their original shapes. Thus, there is no loss of energy in a perfectly elastic collision, and the only energy we have to track is the energy of the objects’ motions.

Now that we have examined the principles and defined our starting conditions, let us go through the math. I will give a brief run-through of the math in the main text. If you want to see the full series of steps, see the attached document at the end of this post.

Using 𝑚𝑣 for momentum and ½𝑚𝑣² for energy, the equation

𝑚₁𝑣_1i+𝑚₂𝑣_2i=𝑚₁𝑣_1f+𝑚₂𝑣_2f

shows us that the initial and final momentums are equal. Similarly, the equation

½𝑚₁𝑣_1i²+½𝑚₂𝑣_2i²=½𝑚₁𝑣_1f²+½𝑚₂𝑣_2f²

𝑚₁𝑣_1i²+𝑚₂𝑣_2i²=𝑚₁𝑣_1f²+𝑚₂𝑣_2f²

shows us that the initial and final energies are equal. Now, we can take the first equation, the momentum equation, and solve for 𝑣_2f. Doing so gives us

Now, we put this into the energy equation and solve for 𝑣_1f. Doing so actually results in a quadratic equation, which is

We can now use the quadratic formula where  𝑚₁ + 𝑚₂ = a, -2(𝑚₁𝑣_1i + 𝑚₂𝑣_2i) = b, and 𝑚₁𝑣_1i + 2𝑚₂𝑣_2i – 𝑚₂𝑣_1i = c. Doing so gives us

Using a similar series of steps, we can also solve for 𝑣_2f, which gives us

Notice that in both equations, everything on the right is a known quantity. We have already specified that the masses and initial velocities of both objects will be known. Since we know the quantities on the right hand, we can solve for both final velocities.

I created an Excel file that has a place to enter the masses and initial velocities of two objects and gives the final velocities in return. The file also shows two graphs, showing what the interaction between these two objects looks like. Note that the first graph shows the objects moving toward one another prior to the collision while the second graph shows the objects moving away from each other, after the collision. Note that the velocities can be either positive or negative, since velocity is a vector. If you want the objects to be moving toward one another, give one object a positive velocity and the other a negative velocity. If you want both objects to move in the same direction, give them both positive velocities. Mass should remain positive since mass is a scalar.

The calculations in the Excel file are rather simple, but they let us predict or explain how objects interact with one another. Consider the following video. Follow the link to see the video.

https://rumble.com/v11n700-demonstrating-a-collision.html

Why does the tennis ball bounce up so high? When both it and the basketball are dropped together, they fall at the same rate. When the basketball hits the ground, its velocity is reversed (since it bounces off the ground) but its speed should remain the same. In other words, it has the same but negative velocity relative to the tennis ball (to make the illustration better, we will actually treat the velocity of the tennis ball as negative and that of the basketball as positive: it makes the graphs look better). The basketball has a greater mass than the tennis ball. Let us say that the basketball has 5 times more mass than the tennis ball. Picking a mass for the tennis ball and a velocity for both balls, we end up with a result like this:

Notice that we successfully predict that the tennis ball bounces considerably higher than the basketball. The Excel file acts as a simulation of real life.

In case you are wondering why the tennis ball bounces so much higher, it is because of the mass difference. If the basketball is 5 times heavier than the tennis ball, the basketball has 5 times the energy of the tennis ball. When they collide, a good portion of that energy is transferred to the tennis ball, causing it to bounce higher.

As I said at the beginning, one of the interesting things about physics is how well it can be used to predict the interactions of objects. As has been demonstrated here, we can predict the outcome of the collision of two objects using some rather simple equations in physics.

Thoughts from Steven

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^[1]Technically, the weight varies with the gravitational field, but the mass will not change.

^[2]Technically, no energy is lost, as the law of conservation of energy tells us. However, the energy is lost in the sense that no longer affects the movement of the balls.