A bicycle and its rider have a total mass of 72 kilograms, and they initially moved at a constant speed of nine meters per second. The bicycle was ridden in a straight line through a puddle for a time of 1.2 seconds and when the bicycle had crossed the puddle, it had a momentum of 540 kilogram-meters per second. What average force was applied to the bicycle by the puddle?
Okay, in this question, we have somebody riding a bicycle. And we’re told that the total mass of the bicycle and its rider is 72 kilograms. We’ll label this mass as 𝑚 so that 𝑚 is equal to 72 kilograms. We are told that our rider on their bicycle initially moved at a speed of nine meters per second and that they then ride in a straight line through a puddle. So, immediately before they go through the puddle, we can say that the velocity of the rider on their bike, which we’ll label 𝑣, is equal to nine meters per second in this direction straight through the puddle.
The question then tells us that they ride through this puddle for a time of 1.2 seconds. We’ll label this time as Δ𝑡 so that Δ𝑡 is equal to 1.2 seconds. Here, the Δ means that we’re measuring a change in the quantity 𝑡 because this value of 1.2 seconds is the amount of time that passes while the cyclist rides through the puddle or, equivalently, the change in time between the moment when the cyclist enters the puddle and the moment when they leave it. The final piece of information that we are given is that when the bicycle had crossed the puddle, it had a momentum of 540 kilogram meters per second. We’ll label this momentum as 𝑝 subscript 𝑓, where the 𝑓 stands for final, since this is the final momentum after traveling through the puddle. So we have that 𝑝 subscript 𝑓 is equal to 540 kilogram meters per second.
We are asked to work out what the average force is that’s applied to the bicycle by the puddle. To calculate this average force, we can recall that Newton’s second law may be expressed to say that the average force on an object is equal to the change in momentum of that object, Δ𝑝, divided by the time over which that momentum changes, Δ𝑡. In terms of the physics of our particular situation, we’re talking about an average force because the actual force may vary in time, for example, if one bit of the puddle is deeper than another.
Mathematically, this expression gives the average force because we’re taking the total momentum change between the moment when the rider enters the puddle and the moment when they leave it and dividing that by the total time spent traveling through the puddle. In other words, this expression is averaging what’s happening over the whole of the motion through the puddle, rather than worrying about what’s happening at every single instant along the way. Now, Δ𝑝, the change in momentum of the rider and their bike as they travel through the puddle, is equal to the momentum at the instant that they leave the puddle, which we’ve already labeled 𝑝 subscript 𝑓, minus the momentum of the instant that they enter, which we’ve labeled 𝑝 subscript 𝑖.
We already know that the value of 𝑝 subscript 𝑓 is equal to 540 kilogram-meters per second. So in order to work out the change in momentum Δ𝑝, we just need to find the value of 𝑝 subscript 𝑖. We can recall that the momentum of an object 𝑝 is equal to its mass 𝑚 multiplied by its velocity 𝑣. In our case, the object is the combined system of the bicycle and its rider. We know that the mass of this object is 72 kilograms and that its velocity at the instant the rider enters the puddle is equal to nine meters per second. So we can say that the initial momentum 𝑝 subscript 𝑖 is equal to the mass of 72 kilograms multiplied by the velocity of nine meters per second. Doing this multiplication gives a result of 648 kilogram-meters per second.
Now that we have values for both 𝑝 subscript 𝑓 and 𝑝 subscript 𝑖, we can substitute those values into this equation to calculate our change in momentum Δ𝑝. Doing this substitution gives us that Δ𝑝 is equal to our final momentum of 540 kilogram-meters per second minus our initial momentum of 648 kilogram-meters per second, which gives Δ𝑝 equals negative 108 kilogram meters per second. The fact that Δ𝑝 is negative tells us that the momentum of the rider and their bike decreases as they pass through the puddle. We now know the value of Δ𝑝, the change in momentum of the rider and their bike as they cross the puddle. And we also know the value of Δ𝑡, the time elapsed while the rider cycles through the puddle.
So if we take these values and substitute them in to this equation here, then we’ll be able to calculate the average force that is applied to the bicycle by the puddle. Doing this substitution gives us that the average force 𝐹 is equal to the change in momentum, that’s negative 108 kilogram-meters per second, divided by the time passed. That’s 1.2 seconds. Evaluating this gives us that 𝐹 is equal to negative 90 kilogram-meters per second squared. The fact that this force is negative means that it’s acting in the direction opposite to the motion of the rider and their bicycle. So on the sketch that we drew, this force would be acting to the left.
Now, the base unit of force is the newton, so it makes sense to express our answer in units of newtons. A newton is equivalent to a kilogram-meter per second squared. And since if we look at our value of 𝐹, we see that it has units of kilogram-meters per second squared, then we can just directly rewrite these units as newtons. When we do this, we get our final answer to the question that the average force applied to the bicycle by the puddle is equal to negative 90 newtons.