A motorcycle moving at a velocity of 20 metres per second along a road drives into a patch of mud that stretches for five metres and continually reduces the motorcycles velocity while the motorcycle drives through it. The velocity of the motorcycle when it has completely driven through the mud is 15 metres per second. What was the average acceleration of the motorcycle while driving through the mud?
Okay, so, in this question, we know that we’ve got a motorcycle initially driving along a road. So, here’s our road and here’s what’s supposed to be our motorcycle. Now we’ve been told that this motorcycle drives through a patch of mud. So, here is our patch of mud. And we’ve been told that this patch of mud is five metres long. And then, after that patch of mud, the road presumably continues.
Now we’ve been told that before entering the patch of mud, the motorcycle was driving along at a velocity of 20 metres per second. Now we’ll say that the motorcycle is travelling towards the right because it needs to go through the patch of mud. And we’ve drawn the patch of mud to the right of the road that it was initially travelling on. Now let’s call the initial velocity of the motorcycle 𝑢. And, of course, 𝑢 is equal to 20 metres per second.
Now as soon as the motorcycle enters the mud, it experiences an acceleration due to the mud. However, this might sound a little bit confusing. Why is the motorcycle experiencing an acceleration? Well, this is because the term acceleration doesn’t necessarily have to mean speeding up. It could also mean slowing down. As long as there is a change in velocity, we can call that an acceleration. But specifically, an acceleration where the object is slowing down can also be called a deceleration, or a negative acceleration.
So, the mud causes the bike to accelerate. But this is a negative acceleration because we can see that, after the motorbike leaves the mud, it has a velocity of 15 metres per second. Because that’s what we’ve been told in the question. Let’s call this new velocity 𝑉.
So, to recap, the motorcycle is coming along at 20 metres per second. And then, as soon as it hits the mud, it starts to decelerate, it’s slowing down, until it reaches the end of the mud. At which point, it continues to move forward with its new velocity of 15 metres per second. So, it’s gone from 20 metres per second here to 15 metres per second here. What we need to do is to find the average acceleration of the motorcycle.
Now the reason we have to find the average acceleration is because while the motorcycle is travelling through the mud, it might not necessarily experience the same deceleration at every point in the mud. There might be some lumps and bumps that cause the motorbike to accelerate slightly more or slightly less. But this is not a problem. We can just assume that the deceleration over the entire mud patch was constant. And we need to find the value of that constant acceleration that would result in a motorbike going from 20 metres per second to 15 metres per second over our five-metre patch. Let’s call that constant average acceleration 𝑎, and we need to find out what the value is.
To do this, we can use one of the kinematic equations. Specifically, we need to use this equation. This equation tells us that the square of the final velocity of an object, in this case, our motorcycle, is equal to its initial velocity squared plus two times its acceleration multiplied by the distance over which it’s accelerating. Now we already know the values of 𝑉, 𝑢, and 𝑠. Because remember, 𝑠 is five metres. That’s the length of the patch of mud over which the motorcycle is accelerating. And so, the only quantity we don’t know in this equation is 𝑎. And that’s exactly what we’re trying to find out.
Now bear in mind, before we do anything else, we need to realise that because the motorcycle is decelerating, in other words, it starts out faster than it finishes, we are expecting the value of 𝑎 to be negative. This is because the mud essentially causes an acceleration to the motorcycle in this direction, in the direction opposite to its motion. And that’s what results in the motorcycle slowing down.
And this ties into the whole idea that acceleration is actually a vector quantity. This means that not only does it have a magnitude, or size, but it also has a direction. And the fact that we’re expecting a negative value for the acceleration will account for that direction. It will tell us that the acceleration of the motorbike is in the opposite direction to the velocity. Because we assume that the velocity travelling this way is in a positive direction.
So, to find the value of 𝑎 using this equation, we need to rearrange first. Let’s start by subtracting 𝑢 squared from both sides. This way, on the right-hand side, we’ve got 𝑢 squared minus 𝑢 squared which cancel. And so, we’re left with 𝑉 squared minus 𝑢 squared on the left and just two 𝑎𝑠 on the right.
Then, we can divide both sides of the equation by two 𝑠 because this way the twos on the right-hand side cancel and the 𝑠’s also cancel. And that leaves us with 𝑉 squared minus 𝑢 squared divided by two 𝑠 on the left and just the acceleration 𝑎 on the right. So, at this point, we’ve rearranged the equation as we need to and we can plug in our values now.
We can say that the acceleration of the motorcycle 𝑎 is equal to 𝑉 squared, that’s the final velocity squared, which is 15 metres per second whole squared, minus 𝑢 squared, that’s the initial velocity whole squared, divided by two times 𝑠, or, in other words, two times five metres. 𝑠 is the length of patch of mud over which the motorcycle decelerates.
Now we can first start by simplifying everything on the numerator. We see that 15 squared is equal to 225. And metres per second whole squared is equal to metres squared divided by second squared. And so, we have for this term, 225 metres squared per second squared. And then, for the second term, we subtract 20 squared, which is 400. And once again, the unit is going to be metres squared per second squared. And so, that’s exactly what we write down for the second term.
Now since both the terms of the numerator have the same units, metres squared per second squared, we can, therefore, subtract 400 from 225. And we can combine both terms into one term, which becomes negative 175 metres squared per second squared. Then, looking at the denominator, we see that two multiplied by five metres is equal to 10 metres. And so, we write that in the denominator. Then, we can divide negative 175 by 10 and we can divide the units metres squared per second squared by metres.
Just looking at the units for now, we see that one power of metres in the numerator cancels with the power of metres in the denominator, leaving us with only metres per second squared as the overall unit. And that’s exactly what we want because we’re looking for an acceleration on the left-hand side. And acceleration has units of metres per second squared.
So, all that’s left to do now is to find out what negative 175 divided by 10 is, which ends up being negative 17.5. And the unit, as we said earlier, is metres per second squared. And since we were asked to find the average acceleration of the motorcycle while driving through the mud, we have, therefore, arrived at our final answer. This acceleration happens to be negative 17.5 metres per second squared.