Suppose the position function for a free falling object on a certain planet is given S(t) = -14t^2 + vot + so. A silver coin is dropped from the top of a building that is 1370feet tall. Find the velocity of the coin on impact.

Since the coin is dropped, its initial velocity is equal to 0. So, to find the velocity, we need to derivate the given function (position). We need to remember that the derivate of the position is the velocity.

So, we have the equation:

[tex]s(t)=-14t^{2}+vo(t)+so\\\\s(t)=-14t^{2} +o+1370[/tex]

Now, derivating, we have:

[tex]f(x)=x^{2} \\f'(x)=2*x^{2-1}=2x\\\\s(t)=-14t^{2}+1370\\\\s'(t)=-14*2*t^{2-1}=-28t[/tex]

So,

[tex]v(t)=-28t[/tex]

Then, to calculate the time taken for the coin to impact (height equal to 0), we have:

[tex]s(t)=-14t^{2}+vo(t)+so\\\\s(t)=-14t^{2}+1370\\\\0=-14t^{2}+1370\\\\14t^{2}=1370\\\\t^{2}=\frac{1370}{14}=97.85\\\\t=+-\sqrt{97.85}=+-9.89[/tex]

Since negative times does not exist, we have that the time taken is equal to 9.89 seconds.

Finally, calculating the speed at the impact, we have:

[tex]v(t)=-28t\\\\v=-28*9.89s=-276.92\frac{ft}{s}[/tex]

Have a nice day!