How to Calculate Momentum

Introduction

This article is going to focus on the conservation of momentum in a closed system using a collision with a lorry and a car and how those two objects are affected.

What is Momentum

Momentum (p) is given when we multiply an object’s mass (m) by the same object’s velocity (v); p =mv. In this case a lorry going at 60mph is going to have a larger momentum than a car going at 60mph.

What is a Closed System

A closed system presumes that the effects of energy transference is negligible. In this case study we are assuming a closed system because it makes learning easier; friction, wind resistance, etc is not included. 

Lorry Vs Car

Here we have two vehicles travelling at 60mph toward each other (let’s assume they’re playing chicken) in a straight line. The lorry has a weight of 3000 kg and the car 1000 kg. In order calculate velocity we need to convert mph into m/s. To do this we need to know that there are 1609 meters per mile so therefore, 60 mph is equal to 96540 meter per hour. To get to meters per second we will need to divide by 3600 as there are that many in an hour: 96540/3600 = 26.82 m/s.

It is important to note that because velocity is a vector quantity one of these values must be negative if there is a head on collision. In these case because the lorry is heading in from left to right it is in the positive direction and the car is in a negative direction because it is travelling from right to left. It doesn’t really matter which vehicle it is we just have to assign these values to be consistent.

The next step is to find the inital momentum for both vehicles; that is as they are now before the collision. 

• Lorry: initial momentum = 3000 kg x 26.82 m/s = 80460 kgm/s
• Car: initial momentum = 1000 kg x -26.82 m/s = -26820 kgm/s

When they collide we have to assert that the collision is an inelastic one as such the masses of both the lorry and the car are added together to create a new mass: 3000 kg + 1000 kg = 4000kg. The conservation of momentum stipulates that the final momentum much be equal to the initial momentum of the object. As the new object is the sum of both the previous initial momentums which would be 80460 kgm/s – 26820 kgm/s, the final momentum is 53640 kgm/s. As we now that the new object’s mass will be 4000 kg we can work out the new object’s velocity through the process of rearranging the formula. 

V = p/m. Thus, v = (53640 kgm/s) / (4000 kg) = 13.41 m/s. 

So if we take these values for our case study we can see that when the collision happened the lorry slowed from 26.82 m/s to 13.41 m/s and the car changed a massive -26.82 m/s to a positive 13.41 m/s so it is actually going in the opposite direction to its initial velocity. 

We can now work out the impact of force (f) on the car at the point of collision assuming it takes place over the course of one second. F = change in momentum (p) / change in time (t). Therefore, change in momentum = 53640 kgm/s – 26820 kgm/s = 26820 kgm/s. Therefore, the force of impact is 26820 N. 

Obviously this is a simplified situation which doesn’t take into account things such as structural crumple zones to absorb force, nor does it allow for the fact that not all of the collision will involve all of the masses of both the lorry and the car. However, working at these “idealised” conditions we can deduce that the limit of such a force impact would 26820 N, in reality it is likely to be less than this impact force due to open system environmental factors and those other things mentioned above. Bear in mind we’re describing a one-dimensional closed system example of what would be a three-dimensional open system event, but it does help to understand the science.

For an interesting and humorous application one-dimensional closed system of momentum check out this article: Bullying with Science Part 1