Video: Changing the Subject of a Formula

The displacement, ๐‘ , of an object traveling with uniform acceleration, ๐‘Ž, can be calculated using the formula ๐‘  = ๐‘ข๐‘ก + (1/2) ๐‘Ž๐‘กยฒ, where ๐‘ข is the initial velocity and ๐‘ก is the time. Make ๐‘Ž the subject. Calculate the acceleration of an object that has started from rest and travelled 200 m in 8 s.

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Video Transcript

The displacement, ๐‘ , of an object travelling with uniform acceleration, ๐‘Ž, can be calculated using the formula ๐‘  equals ๐‘ข๐‘ก plus a half ๐‘Ž๐‘ก squared, where ๐‘ข is the initial velocity and ๐‘ก is the time. Thereโ€™re two parts to this question. The first part, Make ๐‘Ž the subject. And the second part, calculate the acceleration of an object that has started from rest and travelled 200 meters in eight seconds.

So if we have our formula, which is ๐‘  equals ๐‘ข๐‘ก plus half ๐‘Ž๐‘ก squared. And if we want to make ๐‘Ž the subject, well, the subject means we want the ๐‘Ž on its own. So the first thing weโ€™re gonna do is subtract ๐‘ข๐‘ก from both sides of the equation. Thatโ€™s cause that will leave us with a half ๐‘Ž๐‘ก squared on the right-hand side. So now, when we subtracted ๐‘ข๐‘ก from both sides of the equation, we get ๐‘  minus ๐‘ข๐‘ก is equal to a half ๐‘Ž๐‘ก squared. And now, because weโ€™ve got a half ๐‘Ž๐‘ก squared on the right-hand side, what weโ€™re gonna do is multiply both sides of the equation by two. And thatโ€™s because that will remove the half that weโ€™ve got removed the fraction.

So when we do that, we get two multiplied by ๐‘  minus ๐‘ข๐‘ก is equal to ๐‘Ž๐‘ก squared. Thatโ€™s because if we have a half multiplied by two, you get one. So then, all we need to do is divide by ๐‘ก squared because that will leave ๐‘Ž on its own. And once we do that, weโ€™ll have two multiplied by ๐‘  minus ๐‘ข๐‘ก over ๐‘ก squared is equal to ๐‘Ž. So therefore, having made ๐‘Ž the subject, weโ€™ve got ๐‘Ž is equal to two multiplied by ๐‘  minus ๐‘ข๐‘ก over ๐‘ก squared.

Great. Now, letโ€™s move on to the second part. And in the second part, we need to calculate the acceleration of an object that has started from rest and travelled 200 meters in eight seconds. So now to solve any question like this, what Iโ€™d do first is write down the information Iโ€™ve got. So we know that ๐‘ข is equal to zero because this is the initial velocity. And we know that because weโ€™re told that the object starts from rest. And for an object to start from rest, that means that itโ€™s not gonna have any velocity at the beginning. So ๐‘ข is equal to zero.

Then, the next thing we know is that the distance, which is ๐‘ , is equal to 200 meters. And thatโ€™s because weโ€™re told the distance travelled is 200 meters. Then, we also know the time because this is eight seconds and the acceleration ๐‘Ž is what weโ€™re looking for. So we havenโ€™t got that yet. Okay, now weโ€™ve got our values, we can just substitute them into our equation. And the equation weโ€™re gonna use is ๐‘Ž equals two multiplied by ๐‘  minus ๐‘ข๐‘ก over ๐‘ก squared because what weโ€™re trying to find is ๐‘Ž, which is the acceleration.

So now, to work out the acceleration, weโ€™re gonna have ๐‘Ž is equal to two multiplied by. Then, weโ€™ve got 200 cause thatโ€™s our ๐‘ , cause thatโ€™s our distance minus ๐‘ข, which is zero multiplied by our time, which is eight. And then this is all divided by our time squared, which is eight squared. So this gives us that ๐‘Ž is equal to 400 divided by 64. And thatโ€™s because we had two multiplied by. And then, youโ€™ve got 200 minus zero multiplied by eight. Well, 200 minus zero cause zero multiplied by anything is just zero gives us 200. Two lots of 200 is 400 and then divided by eight squared and eight squared is 64.

Then, if we simplified it by dividing both the numerator and denominator by four, weโ€™re gonna get 100 over 16. And then once more divided by four, weโ€™re gonna get 25 over four cause 100 divided by four is 25. 16 divided by four is four. So then, we can see that four goes into 25 six times with one remainder. So we get six and a quarter. And when we turn it into a decimal, itโ€™s 6.25. Then, we know that the units are gonna be meters per second squared because thatโ€™s the measurement weโ€™re gonna use for acceleration.

So we can say that if we make ๐‘Ž the subject, we get ๐‘Ž equals two multiplied by ๐‘  minus ๐‘ข๐‘ก over ๐‘ก squared. And if we calculated the acceleration of an object that has started from rest and travelled 200 meters in eight seconds, then the result is 6.25 meters per second.

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