### Video Transcript

A bullet fired from a gun leaves the end of the 55-centimeter-long gun barrel at a speed of 500 meters per second and hits a bottle 35 meters away, as shown in the diagram. What is the average acceleration of the bullet in the gun barrel? Round your answer to the nearest kilometer per square second.

Okay, so in this question, we are asked to work out the average acceleration of a bullet fired from a gun. The bullet is only accelerating while itโs in the gun barrel, so the distance over which it accelerates is 55 centimeters. We are told that the bullet leaves the gun at the end of this 55-centimeter distance with a speed of 500 meters per second. Before the gun is fired, we know that the bullet will be at rest, so we can say that it starts with an initial speed of zero meters per second.

Weโll label this initial speed as ๐ข, and weโll label the bulletโs final speed of 500 meters per second as ๐ฃ. While weโre at it, letโs label the 55-centimeter length of the gun barrel as ๐ . So the bullet accelerates from an initial speed of ๐ข equal to zero meters per second to a final speed of ๐ฃ equal to 500 meters per second over a distance of ๐ equal to 55 centimeters. We are trying to work out the average acceleration of the bullet. So letโs label this average acceleration as ๐.

Now, it turns out that thereโs an equation that links an objectโs initial speed ๐ข, its final speed ๐ฃ, its acceleration ๐, and the distance ๐ over which it accelerates. Specifically, that equation tells us that ๐ฃ squared is equal to ๐ข squared plus two times ๐ times ๐ . In this case, we know the values of the quantities ๐ฃ, ๐ข, and ๐ . Weโre trying to work out the acceleration ๐. This means that we want to take this equation and rearrange it to make ๐ the subject.

The first step is to subtract ๐ข squared from both sides of the equation. Then, on the right-hand side, we have ๐ข squared, which cancels out with the minus ๐ข squared. This leaves us with an equation that says ๐ฃ squared minus ๐ข squared is equal to two times ๐ times ๐ . Then, we divide both sides of the equation by two ๐ . On the right-hand side, the two in the numerator cancels with the two in the denominator and the ๐ in the numerator cancels with the ๐ in the denominator. This gives us an equation that says ๐ฃ squared minus ๐ข squared divided by two ๐ is equal to ๐. And of course, we can also write this equation the other way around to say that ๐ is equal to ๐ฃ squared minus ๐ข squared divided by two ๐ .

So we now have an equation that will allow us to calculate the value of ๐ as long as we know the values of ๐ฃ, ๐ข, and ๐ .

If we look back at the question, we see that we are asked to give our answer for the acceleration to the nearest kilometer per square second. However, at the moment our speeds are in units of meters per second and our distance is in units of centimeters. We need to convert the two speeds to units of kilometers per second and the distance to units of kilometers.

In order to do this, we need to recall that one kilometer is equal to 1000 meters and one meter is equal to 100 centimeters. Now, if one kilometer is equal to 1000 meters, then one meter must be equal to one thousandth of a kilometer. Likewise, if one meter is equal to 100 centimeters, then one centimeter is one hundredth of a meter. So to convert the distance ๐ of 55 centimeters into units of kilometers, we take the value in units of centimeters and we multiply it by one over 100 meters per centimeter, then again by one over 1000 kilometers per meter.

Looking at the units, we can see that the centimeters cancel with the per centimeter and the meters cancel with the per meter. So we are left with units of kilometers. Evaluating the expression, we find that ๐ is equal to 0.00055 kilometers. Now, we just need to convert our speeds to units of kilometers per second. Letโs clear ourselves some space to do this. Weโll start with our speed of ๐ฃ equal to 500 meters per second. To get this speed into units of kilometers per second, we need to multiply the value in meters per second by one over 1000 kilometers per meter.

Then, in terms of the units, the meters cancel with the per meter, and we are left with units of kilometers per second. When we evaluate this expression, we find that ๐ฃ, the speed of the bullet when it leaves the gun barrel, is equal to 0.5 kilometers per second. Now, we could also do the same thing for the initial speed ๐ข. But itโs easier to just notice that since this initial speed is zero meters per second, then itโs going to be zero in whatever units we choose to express it in. After all, if something isnโt moving, then, well, itโs not moving, no matter what units we choose to measure that lack of movement in.

So we know that ๐ข is equal to zero kilometers per second. And weโve also worked out that ๐ฃ is equal to 0.5 kilometers per second. So we now have values for ๐ , ๐ข, and ๐ฃ in units of kilometers and kilometers per second. All thatโs left for us to do is to sub those values into this equation to calculate the value of ๐.

Letโs clear some space so that we can do this. We know that ๐ is equal to ๐ฃ squared minus ๐ข squared divided by two ๐ . In place of ๐ฃ, weโll sub in our value of 0.5 kilometers per second. And in place of ๐ข, weโll sub in the value of zero kilometers per second. Finally, in place of the ๐ in the denominator, weโll sub in our value of 0.00055 kilometers. This gives us an expression for ๐ and all that we need to do now is to evaluate it.

In the numerator, 0.5 kilometers per second all squared gives us 0.25 kilometers squared per second squared. And zero kilometers per second all squared is zero kilometers squared per second squared. In the denominator, two multiplied by 0.00055 kilometers gives us 0.0011 kilometers. In the numerator of this expression, we have 0.25 kilometers squared per second squared minus zero kilometers squared per second squared. And this is simply equal to 0.25 kilometers squared per second squared.

In terms of the units, one factor of the kilometers from the numerator cancels with the kilometers from the denominator. Then, evaluating the expression gives us that the acceleration ๐ is equal to 227.27 kilometers per second squared, where the bar over these two digits indicates that they are recurring. Finally, we should notice that the question asks us to round our answer to the nearest kilometer per square second. So we need to take this value that weโve calculated for ๐ and round it to the nearest whole number of kilometers per second squared. When we do this, we get our answer to this first part of the question that, to the nearest kilometer per square second, the average acceleration of the bullet in the gun barrel is equal to 227 kilometers per second squared.

Okay, now letโs look at the second part of the question.

How much time after the bullet leaves the gun does it hit the bottle, assuming it moves with a constant velocity?

Okay, so we know that the bullet leaves the gun with a speed of 500 meters per second. And we can see from the diagram that the bullet is headed directly toward the bottle. We are told to assume that the bullet moves with a constant velocity. This means that we can assume that both its speed and its direction remain constant. We are told that the bottle is 35 meters away from the gun. So we know that the bullet moves a distance of 35 meters at a speed of 500 meters per second.

We are asked to work out how much time it takes for the bullet to move this 35-meter distance from the gun to the bottle. Weโll label this distance as ๐ and the time that weโre trying to calculate as ๐ก, and weโve already labeled the bulletโs speed as ๐ฃ. We can recall that these three quantities are linked by the equation ๐ฃ is equal to ๐ divided by ๐ก. Or in words, for an object moving with a constant speed, that speed is equal to the distance moved by the object divided by the time taken to move the distance.

In this case, we know the values of ๐ฃ and ๐, and weโre trying to work out the value of the quantity ๐ก. So letโs take this equation and rearrange it to make ๐ก the subject. To do this, the first step is to multiply both sides of the equation by ๐ก. Then on the right-hand side, the ๐ก in the numerator cancels with the ๐ก in the denominator. This leaves us with an equation that says ๐ฃ multiplied by ๐ก is equal to ๐. Then, we divide both sides of the equation by ๐ฃ. On the left-hand side, the ๐ฃ in the numerator cancels with the ๐ฃ in the denominator.

So we end up with an equation that says time ๐ก is equal to distance ๐ divided by speed ๐ฃ. Now, we just need to sub our values for ๐ and ๐ฃ into this equation. When we do this, we get that ๐ก is equal to 35 meters, thatโs our value for ๐, divided by 500 meters per second, thatโs our value for ๐ฃ. Evaluating this expression, we find that ๐ก is equal to 0.07 seconds. And so our answer to this second part of the question is that the bullet hits the bottle 0.07 seconds after it leaves the gun.