You are running towards another player to meet in a tackle in a game of rugby. How can you ensure that you are not pushed backwards in the collision that is about to take place?
By the end of this chapter you should be able to:
•Explain the concept of conservation of momentum in the context of collisions
•Predict the outcome of collisions if the bodies’ masses and velocities are known
•Use this information to improve the outcome of a collision for a player or athlete
Remember from Chapter 8 that the Law of Conservation of Momentum states that the momentum of a system remains unchanged unless it is acted upon by an external force. In a collision, the total momentum of the system is equal to the sum of the mass × velocity of all the colliding objects; that is, momentum = m1v1 + m2v2 ... From this equation, you can see that it is easy to work out what might happen in a collision.
Let’s say you have a mass of 80 kg and your opponent has a mass of 100 kg. You are moving towards your opponent at 2 m·s-1 and your opponent is running at you at 5 m·s-1. What will happen when you collide? The total momentum (ptot) of the system must remain the same. Currently, the combined momentum is:
ptot = 100 kg × 5 m·s-1 + 80 kg × 2 m·s-1
= 500 + 160 = 660 kg·m·s-1
The momentum must remain constant after the collision but how will the players be moving?
Before collision |
After collision |
m1v1 + m2v2 |
= m1v1 + m2v2 |
m1v1 + m2v2 |
= (m1 + m2)v ... the players move together at one velocity |
(100 × 5) + (80 × -2) |
= (100 + 80) × v |
(v2 is -2 m·s-1 because the players are running in opposite directions) |
|
340 |
= 180 × v |
Dividing both sides by 180: |
|
340 / 180 = v |
|
1.8 m·s-1 = v |
|
So the two players will be moving at 1.8 m·s-1 after the collision. Since the value is positive, it means that they will move in the direction of the player whose velocity was positive (the 100 kg player) and the 80 kg player will be forced backwards.
THE ANSWER
How can you make sure you continue to move forwards in such a collision as shown in Figure 10.1? You must have a greater momentum going into the collision. Since your body mass is smaller, you’d have to have a greater velocity. We can work out the velocity at which you would exactly match your opposition and the velocity above which you would knock your opponent backwards. Your velocity is represented by v2, so we need to re-arrange the equation to calculate this number with the total final velocity of the system (v) at zero. I’ve written it out step-by-step below.
Before collision |
After collision |
m1v1 + m2v2 |
= m1v1 + m2v2 |
m1v1 + m2v2 |
= (m1 + m2)v |
m1v1 + m2v2 |
= 0 (since v = 0) |
m1v1 |
= -m2v2 |
m1v1 / -m2 |
= v2 |
100 kg × 5 m·s-1 / -80 kg |
= v2 |
-6.25 m·s-1 |
= v2 |
So, if you were to run towards your opponent at 6.25 m·s-1, you would have a resulting velocity of zero after the collision. If you run more quickly, your opponent would be pushed backwards.
Actually, there is a slightly easier way to do this. If you both had the same momentum as you collided, your velocities after the collision would be zero. So you could calculate your opponent’s momentum (500 kg·m·s-1) and then find out what velocity you’d need to run at, given that your mass is 80 kg, to have the same momentum (m1v1 = 500 kg·m·s-1, so v2 = 500 kg / 80 kg·m·s-1 = 6.25 m·s-1).
There is another way to ensure your opponent is pushed backwards that doesn’t require you to run at breakneck speed. Remember that the total momentum of the system must remain the same, because momentum is conserved unless an external force acts. So a second way to make the opponent move backwards is to continue to apply a force to the ground during the collision so that the ground applies an equal and opposite force back at you! You are, in effect, performing work on your opponent during the collision. To do this, you need to apply the force with your legs while your upper body absorbs the force (or shock) of the impact. So when your coach says ‘drive into your opponent’, that’s what they mean.
FIG. 10.1 While the momentum of an ‘ideal’ collision is constant, we can apply an external force during the collision in order to push back an object (e.g. an opposing player) that had a greater initial momentum.
HOW ELSE CAN WE USE THIS INFORMATION?
Remember that velocity is a vector quantity, meaning it is described by a magnitude and a direction; so momentum is also a vector quantity. You might have a fast player with a large mass (that is, a high momentum) running towards you, which means you need to oppose them with a large momentum of your own. Or not. If you step to one side and let the player run to the side of you before you attempt the tackle, their velocity, and therefore their momentum, in your direction is effectively zero. This is shown in Figure 10.1. Since the component of the velocity, and therefore of the momentum, directed at you is zero, you only need to tackle them with a small momentum to win in the collision.
As a general rule, if we understand what will happen in a ‘normal’ collision, we can work out what will happen when any two objects collide. For example, we could work out how fast a ball will travel after it makes contact with a moving bat, as you will see in the next chapter. We can also understand why we should ‘give’ with the ball when we catch (Figure 10.2).
FIG. 10.2 Catching a ball is made easier when the hands move at a velocity in the same direction as the oncoming ball, but with slightly lesser magnitude. The lower resultant impact velocity slows the velocity at which the ball would rebound in the collision with the hands, which makes it easier to time the clasping of the hands.
A ball coming to us at a high speed (we’ll call it a positive speed, since it is coming towards us; a negative speed would indicate that the ball is moving away) has a high momentum but after a collision with our stationary hands it will leave with the same velocity but in the negative direction. This makes the ball hard to catch because we’d have to close our hands at precisely the right moment to stop the fast-moving ball rebounding. The high-speed impact might also hurt a lot! If we move our hands with a positive velocity (that is, in the direction of the ball) then the relative velocity of the ball impact is lower and the ball will tend to rebound with a lower velocity. We have more time to close our hands and prevent the ball from rebounding away. It will also hurt less, since the impact velocity is lower.
Useful Equations
momentum (p) = m × v
conservation of momentum: m1v1 = m2v2
impulse (J) = F × t or ∆mv
Related Websites
Hyperphysics (http://hyperphysics.phy-astr.gsu.edu/hbase/elacol.html). Descriptions, movies and examples of elastic and inelastic collisions.
The Science House (http://www.science-house.org/student/bw/sports/collision.html). Description and activities relating to collisions in sports.
The Physics of Sports (http://www.topendsports.com/biomechanics/physics.htm). Website investigating the applications of physics in sports.
The Physics Classroom – Multimedia tools (http://www.physicsclassroom.com/mmedia/). Interactive tools and movies depicting basic physics concepts
