### Video Transcript

A particle starts from rest and moves in a straight line. At a time ๐ก seconds, its velocity is given by ๐ฃ is equal to 36๐ก minus 45๐ก squared centimeters per second, where ๐ก is greater than or equal to zero. How long does it take the particle to return to its starting place?

Weโre given the velocity function of a particle which is moving in a straight line. And this velocity function is only valid for values of ๐ก greater than or equal to zero. Our particle starts from rest. And we need to work out how long it takes our particle to return to its starting place. The first question we should ask is, what is our particleโs starting place?

Weโll call the displacement function of our particle after ๐ก seconds ๐ of ๐ก. Then, its starting position is just ๐ evaluated at ๐ก is equal to zero. Now, we notice if ๐ of ๐ก is equal to ๐ of zero, then our particle is at our starting place. So, we just need to solve ๐ of zero is equal to ๐ of ๐ก, where ๐ก is not equal to zero.

So, weโll need to find an expression for ๐ of ๐ก. Weโll start by recalling that the velocity of our particle is the rate of change of displacement with respect to time. If this is true, we can then integrate both sides of this equation with respect to ๐ก. Weโll get the integral of our velocity function with respect to ๐ก is equal to our displacement function. And weโre given the velocity function of our particle. So, weโll use this to find an expression for our displacement function ๐ of ๐ก.

Itโs equal to the integral of 36๐ก minus 45๐ก squared with respect to ๐ก. We can then integrate this term by term by using the power rule for integration. We want to add one to our exponent of ๐ก and then divide by this new exponent of ๐ก. This gives us ๐ of ๐ก is equal to 18๐ก squared minus 15๐ก cubed plus our constant of integration ๐ถ.

Remember, weโre trying to find out how long it takes our particle to return to its starting place. So, letโs find its starting place. Weโll substitute ๐ก is equal to zero into our expression for ๐ of ๐ก. Substituting ๐ก is equal to zero, we get the initial displacement of our particle is equal to 18 times zero squared minus 15 times zero cubed plus ๐ถ. And 18 times zero cubed minus 15 times zero squared is equal to zero. So, in actual fact, the initial displacement of our particle is equal to our constant of integration ๐ถ.

At this point, we might be worried. We donโt know how to find the value of ๐ถ in this case. But, in actual fact, we donโt need to find the value of ๐ถ. We just need to solve the equation ๐ of zero is equal to ๐ of ๐ก where ๐ก is not equal to zero. And, of course, since our velocity function was only valid when ๐ก is greater than or equal to zero, our displacement function will also only be valid when ๐ก is greater than or equal to zero.

So, we just need to solve the equation ๐ of zero is equal to ๐ of ๐ก where ๐ก is greater than or equal to zero. First, we showed that ๐ of zero is equal to our constant of integration ๐ถ. Next, we found an expression for ๐ of ๐ก. Itโs equal to 18๐ก squared minus 15๐ก cubed plus ๐ถ. And we want to solve this equation. Weโll start by subtracting ๐ถ from both sides of the equation. This gives us zero is equal to 18๐ก squared minus 15๐ก cubed.

And to solve this, weโll want to fully factor the right-hand side of this equation. Weโll take out a factor of three and a factor of ๐ก squared. Taking out our factor of three ๐ก squared, we get zero is equal to three ๐ก squared times six minus five ๐ก. So, now, weโre solving an equation where the product of factors is equal to zero. This means one of our factors must be equal to zero. If three ๐ก squared is equal to zero, then ๐ก must be equal to zero.

But remember, weโre not interested in the solution where ๐ก is equal to zero. So, we must have our other factor of six minus five ๐ก is equal to zero. And we can solve this equation for ๐ก, we get that ๐ก is equal to six divided by five. And weโll write this in its decimal form of 1.2. And since all of our units were given in terms of centimeters and seconds, we can give this the units of 1.2 seconds.

Therefore, weโve shown if a particle starts from rest and moves in a straight line. And at a time ๐ก second, its velocity is given by ๐ฃ is equal to 36๐ก minus 45๐ก squared centimeters per second, where ๐ก is greater than or equal to zero. Then after 1.2 seconds, our particle will return to its starting place.