An object that has a momentum of 12 kilogram-meters per second is acted on by a constant force for eight seconds, after which the object’s momentum is six kilogram- meters per second. Determine the force that acted on the object.
Okay, so we have an object that starts out with a momentum of 12 kilogram-meters per second. Let’s imagine that this pink blob here is our object. And we’ll label its initial momentum as 𝐩 one so that we have 𝐩 one is equal to 12 kilogram-meters per second. Now, momentum is a vector quantity, which means that it has a direction as well as a magnitude. Let’s imagine that the object’s momentum is in this direction. We are then told that the object is acted on by a constant force for eight seconds and that after this eight-second interval, the object’s momentum is six kilogram-meters per second. So we can draw our object again after a time interval of eight seconds, which we’ve labeled as Δ𝑡. We’ll label its new momentum as 𝐩 two so that we have 𝐩 two is equal to six kilogram-meters per second.
Since the original momentum 𝐩 one was positive and the new momentum 𝐩 two is also positive, then 𝐩 two will be in the same direction as 𝐩 one. So we can represent that with an arrow like this. We are asked to work out the force which acted on the object to cause this change in momentum. We can recall that whenever we have a force which acts to change the momentum of an object, then if the object’s momentum changes by an amount of Δ𝐩 over a time interval of Δ𝑡, then the force 𝐹 which acts on that object is equal to Δ𝐩 divided by Δ𝑡. So we know the time interval Δ𝑡, and we know the momentum the object started with and the momentum it ended up with.
We can use these values of 𝐩 one and 𝐩 two to calculate the change in the object’s momentum Δ𝐩. Δ𝐩 must be equal to the final momentum 𝐩 two minus the initial momentum 𝐩 one. If we now sub in the values for 𝐩 two and 𝐩 one, we find that the change in momentum is equal to negative six kilogram-meters per second. The value we’ve calculated for Δ𝐩 is negative because 𝐩 two is smaller than 𝐩 one. In other words, the momentum of the object has decreased, and so we have a negative change in the object’s momentum. We now have values for both Δ𝐩 and Δ𝑡. So it’s time to go ahead and sub those values into this equation to calculate the force 𝐹. When we do this, we get that 𝐹 is equal to negative six kilogram-meters per second divided by eight seconds.
At this stage, it’s worth pointing out that the kilogram-meter per second is the SI base unit for momentum, and the second is the SI base unit for time. So Δ𝐩 and Δ𝑡 are both expressed in their SI base units. This means that the force 𝐹 that we calculate will be in its own SI base unit, which is the newton. When we evaluate this expression for 𝐹, we get a result of negative 0.75 newtons. The fact that this force is negative means that it acts in the opposite direction to the object’s momentum. So since in our diagram we had the momentum of the object directed to the right, then in this diagram, the force will be directed to the left.
This force 𝐹 is exactly what we were asked to calculate. And so our answer to the question is that the force which acted on the object is equal to negative 0.75 newtons.