# Quantifying impulse and force

#### Prerequisites

In our description of the physics behind Newton's laws, we have described the influence that external agents exert on an object in a time interval $Δt$ as an Impulse, $ΔI$, and have implicitly defined it by the equation

$ΔI/m = Δv$.  (During a time interval $Δt$ when an unbalanced influence is acting)

To make sense of this, we have to find some way to quantify and measure the impulse.

Our physical sense of this is that something is pushing or pulling on the object and that's our "external influence."  A very good way to quantify something is to find an object that changes itself in a way that we can measure when the effect we want is present.

As an analogy, think about a thermometer.  When the thermometer is in a situation that our senses would call "hot", the liquid in the thermometer expands, taking up more space.  When the thermometer is in a situation that our senses would call "cold," the liquid in the thermometer contracts, taking up less space.  By putting the liquid in a thin tube, we magnify this effect and use a measurement of the amount of length of the liquid as a measure of how hot or cold something is.  We do this by defining a "standard thermometer."  Somebody (e.g., the National Institute of Standards and Technology), has to keep a standard and send out calibrated thermometers for thermometer manufacturers to compare to in order to make sure everybody is measuring the same thing.

We need to do something like that for impulse.  The core idea for converting our physical sense of push and pull to something measurable is a spring. When we hold the spring in our hands and pull in both directions, the spring changes its length. The harder we pull, the longer it gets. We can feel the spring pulling on our hands, and we can feel that it is pulling harder as it stretches more. So we can create a "standard spring" whose stretch measures how hard it is pulling.

Now let's imagine a simple situation: a standard cart with small wheels (so the rolling of the wheels can be expected to have a negligible effect) being pulled by a spring.

If we pull so that the spring stays stretch by a fixed amount, we know that it pulls on both sides by a fixed amount.  Since we are assuming object egotism, the cart only feels an unbalanced influence of the spring so we can see what the effect is of a constant pull.

I tried to do this in the video at the left. It's not so easy since the cart moves so you have to keep moving back to keep the spring stretched the same amount. To keep the same stretch, I taped a ruler to the cart extending to the front just enough that the spring would stretch to pull the cart, but not too fast. I then moved to try to keep the front end of the spring at the edge of the ruler, so that the spring would always be stretched by the same amount.

You can do this yourself by downloading the video by clicking on the image below and imbedding this movie in a video capture program such as ImageJ or LoggerPro™.

I  used this video to take the data in LoggerPro shown below. What we see is that the velocity keeps changing at a constant rate.  This suggests that the stretch of the spring is not measuring the impulse that the spring delivers, but something else, and that the impulse it delivers is proportional to the amount of time that the influence is applied.  (Why do you think there is a small but consistent oscillation about the linearly increasing velocity?)

We will call the thing that the stretch of the spring is measuring, force. Our observations of the fact that the cart continues to accelerate as long as the stretched spring is pulling on it lead us to define the impulse that the spring delivers as a constant times the time interval. (The change in velocity is proportional to how long a time the cart has been pulled.)

$$FΔt = ΔI$$

This then leads to the equation

$$FΔt/m = Δv$$

or

$$ΔI = FΔt = mΔv$$

This is the critical equation: a force (such as that delivered by a spring that has a constant stretch) acting over a time interval delivers an impulse which changes the mass times the velocity of the object.

We actually should put in a constant, since our standard spring will define a system of units.  But since the equation above is so well established, we can choose our system of units so that we don't have to add a new dimensionality corresponding to force.  We would have to do this if we were introducing a new arbitrary measurement scale.  Instead, we'll look at the equation we obtained and define force so that there is no conversion constant.  Rather, we will write

$$F= m \frac{Δv}{Δt} = ma$$

and we will define a force of "1 unit of force" to be

One unit of force is that force that produces an acceleration of 1 m/s2 when applied to a mass of 1 kg. We will call this 1 Newton = 1 kg-m/s2.

Now what we have to do is figure out what forces there are, how to figure out what they are, and how to put them together.

Joe Redish 9/17/11

Article 345