What Does the Area Under a Force vs Time Graph Represent?

[Tylemaker]

1. The problem
I have trouble understanding slope and area on graphs. For a force as a function of time graph the area is equal to the impulse. Why is that? And what is the slope equal to? How do I know?2. Homework Equations
p=mv
fΔt=mΔv
3. The Attempt at a Solution
So the slope should be equal to f/t?
What on Earth is force divided by time equal to?
And is the area underneath the graph always equal to y*x (in this case y=force, x=time)? So area = f*t = impulse?

 

[ehild]

1. The problem
I have trouble understanding slope and area on graphs. For a force as a function of time graph the area is equal to the impulse. Why is that? And what is the slope equal to? How do I know?

The impulse of a force is defined as I=FΔt for a force F acting for a very short time. And the change of momentum p=mv is equal to the impulse of the force. I=Δp=mv_f-mv_i.

See the first picture: a constant force is plotted. Its impulse is FΔt= F(t_f-t_i), the area under the line F(t) between the initial and final times.

The second picture shows a force which grows linearly with time. Its impulse is equal to the area under the line, which is the same as the average force multiplied by the elapsed time: I=F_avΔt=F_av(t_f-t_i).
The picture on the right shows a general force-time plot. Again, the impulse of the force between t_i and t_f is equal to the area under the curved line.

If you know the impulse of a force imparted to a body, it is equal to the change of momentum. I=mv_f-mv_i.

2. Homework Equations
p=mv
fΔt=mΔv

"f " means the average force during the time period Δt.

3. The Attempt at a Solution
So the slope should be equal to f/t?
What on Earth is force divided by time equal to?
And is the area underneath the graph always equal to y*x (in this case y=force, x=time)? So area = f*t = impulse?

The slope of the f(t) graph is irrelevant. It is not f/t, but the limit of Δf/Δt when Δt get shorter and shorter. (If you study Calculus you will learn that it is the time derivative of f(t).)

The area is fΔt in case f is constant. Otherwise you slice the whole area and add the small pieces f(t)Δt together. You will learn that the area is equal to the integral of f(t) between t_i and t_f.

ehild