When analyzing collisions, it is important to understand how energy is transferred between colliding objects. When **one object collides** with another object, some of the energy of the *first object* is transferred to the *second object*.

How much energy is transferred depends on how the collisions happen. If the collisions are perfect, with no loss of energy, then the **second object would keep moving** at the same speed as before.

However, in reality, there is some degree of loss of energy. This results in both objects slowing down after the collision. The difference in speed before and after the collision is what we call impulse.

Impulse is defined as the change in velocity (speed) of an object as a result of a force acting on it for a duration of time.

## Large collision

Two collision graphs are shown side-by-side above. In each case, the ball is initially at rest, then it is propelled by the force of the bat for a certain time until it comes to a stop.

The difference between the two cases is how long the ball is in motion. In Case A, the ball is in motion for a shorter time, and in Case B, the ball is in motion for a longer time.

You can see that the area under both curves is equal. This means that if you **divide one curve** by the other, you will get the same ratio of area to time.

We can therefore use our knowledge of area under a curve to **determine whether one collision** has more impulse than another simply by looking at how long each **collision occurs**. If they both have the same area under the curve, then they have the same impulse.

## Small collision

When two bodies collide, there is always a change in momentum. If the bodies are static, then the only momentum that changes is the momentum of the body that is moved.

If one body is static and the other body collides with it, then both bodies experience a change in velocity. The moving body experiences a change in velocity in accordance to how it collided with the static body.

The difference between these *two situations reveals something important* about collisions: when * two objects collide*, their masses do not necessarily undergo an equal amount of change in velocity!

In other words, when two objects collide, *one object could experience* a greater (or lesser) change in velocity than the other object. This fact is very interesting; let’s investigate it further.

## Mass of ball 1 = 10 kg

Now let’s look at the second example. Ball 2 is twice the mass of ball 1, so it takes twice as long to accelerate to the same speed as ball 1. When they collide, they stick together, so their velocities are the same after the collision.

The force on ball 2 is opposite the direction of motion, so it acts to slow down ball 2. Because it **takes longer** to accelerate and because of the *stronger force acting* against it, ball 2 has a larger impulse than ball 1.

The last case is a little more complicated. Ball 3 is twice as heavy as ball 1, but it accelerates twice as fast, so its time to collision is equal to that of ball 1. Because it accelerates faster, it has a **higher velocity** after the collision, and thus a higher impulse.

These examples show how *important understanding velocity* and acceleration are when analyzing collisions.

## Mass of ball 2 = 1 kg

When comparing the interactions of ball 1 with ball 2 and ball 2 with ball 1, the mass of the second ball does not matter. Only the speed of the **second ball changes** how the interaction happens.

When **ball 2 hits** the floor, it sinks into the floor until it reaches a certain point of resistance. At that point, it rebounds with a certain amount of momentum based on its mass and how far it sank into the floor.

The same thing happens when **ball 2 hits ball 1**. The difference is that instead of bouncing off the floor, it bounces off another object with similar mass. This changes how much momentum is transferred to the other object, but not whether it occurs.

The Force vs Time graphs for these collisions are shown below. As you can see, there is no significant difference between these graphs.

## Find the speed of the ball after the collision for both cases

Now that you can calculate the speed of the ball after the collision, you can find the difference in speed between the ball and wall. The greater this difference in speed, the greater the impulse of the ball off of the wall.

By comparing these differences in speed, you can determine which collision has the largest impulse. The collision where the ball stops moving has the greatest change in speed, so it has the greatest impulse.

Impulse is measured in joules, a unit of work. One joule is equal to one Newton multiplied by one meter. Since there are newtons in a kilogram-meter (kg-m**), one joule** is equal to * one newton multiplied* by one kilogram-meter (N·kg-m).

The greater the mass of an object, the greater its momentum is. Similarly, objects with higher velocities have higher momentums.

## Use the equation F=mv^{2}/2 to find the impulse for each case

A force-time graph is a way of showing both the force and time components of a motion situation.

It does this by plotting the velocity against time, so that velocity is always increasing or decreasing, depending on the direction.

By doing this, you can see how long it takes to reach a certain velocity, and how long it takes to decrease or increase in velocity. You can also see how **much velocity changes due** to a certain force.

You can also do some *pretty cool physics tricks* with these graphs! For example, you can find the average acceleration by taking the slope of the line connecting the *two points* on the graph that *represent velocities*. You can also find the total time for an event by measuring the length of the line on the graph that represents time.

## The larger collision has the larger impulse

As shown in the above graph, collision A has the **largest impulse**. This is due to the fact that the force is applied for a *longer period* of time. The force is *also stronger*, which increases the **overall impulse**.

## The smaller collision has the larger impulse

In physics, impulse is the integral of force over time. In simpler terms, impulse is the amount of change caused by a force over the time it takes to apply that change.

Impulse is most easily understood through physics collisions. When *two objects collide*, there is a change in momentum. The *collision causes one object* to lose momentum (receives a force) and the other object to gain momentum (provides a force).

A Force vs. Time graph can be constructed for both collisions. The A-B portion of the **blue graph represents** the force applied during the collision, and the length of this line represents how long it applies this force. The **red graph shows** the resulting loss of momentum due to this collision.

As you can see, there is less red in the second half of the graph, meaning that there is less loss of momentum in this half due to this collision.