# Question Video: Calculating the Magnitude of the Impulse on a Body given Its Force–Time Graph Mathematics

The figure shows a force–time graph for a force acting in a constant direction on a body moving along a smooth horizontal plane. Using the information provided, calculate the magnitude of the force’s impulse.

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### Video Transcript

The given figure shows a force–time graph for a force acting in a constant direction on a body moving along a smooth horizontal plane. Using the information provided, calculate the magnitude of the force’s impulse.

In this scenario then, we have a smooth horizontal plane and a body moving across it. The force that this body experiences is plotted against time in our graph. At the outset, there’s zero force. But then, over the next 20 seconds, the force applied to the body increases at a constant rate until it reaches 90 newtons. This force is held constant for the next 50 seconds up until the 70-second mark. At that point, the applied force begins to decrease at a constant rate until at a time of 80 seconds it returns to zero.

With this in mind, we want to calculate the magnitude of the impulse experienced by this body. We can clear a bit of space to work and then recall that the impulse experienced by an object is equal to the net force acting on the object times the time over which that force acts. In our graph, we’re shown the net force acting on our body over time. So to calculate the impulse our body experiences, we’ll multiply each individual force value by the corresponding time interval for that force. This means that the total impulse is equal to the area under this force-versus-time curve.

If we calculate that area, then we’ll have solved for 𝐼. To make this easier, we can divide up this overall area into common shapes. We see that from zero to 20 seconds, this area is a right triangle. Then, from 20 to 70 seconds, it’s a rectangle, and then from 70 to 80 seconds another right triangle. If we call these areas 𝐴 one, 𝐴 two, and 𝐴 three, then the impulse magnitude we want to solve for is actually equal to their sum.

We can start by calculating 𝐴 one. This is the area of a right triangle, which we can recall is equal to one-half the triangle’s base times its height. The base of 𝐴 one we can see from our graph is 20 seconds, and its height we can also see is 90 newtons. This is equal in total to 900 newton seconds, which we then substitute in for 𝐴 one.

Knowing this, we’ll move on to calculate 𝐴 two. This is the area of a rectangle, which is the base of that rectangle multiplied by its height. The rectangle’s base is 70 seconds minus 20 seconds, or 50 seconds, and its height is 90 newtons. This means that 𝐴 two is 4,500 newton seconds, which we can then substitute in to our equation for 𝐼.

Then, finally, we’ll calculate 𝐴 three. This is the area of a right triangle whose base is 80 seconds minus 70 seconds, or 10 seconds, and its height is once again 90 newtons. 𝐴 three then comes out to 450 newton seconds.

We can now calculate 𝐼. And just as a side note, notice that we could have treated the area under our curve as one single trapezoid and solved for it that way. In any case, when we compute the impulse, we find a result of 5,850 newton seconds. This is the magnitude of the impulse experienced by our body.