Roller Coasters

#122 / Dynamics Easy

Consider a roller coaster on a section of a track that is in a straight line. It travels a distance (D). After a certain amount of time (t), D = 10t^4-t^3. Calculate the roller coaster's velocity when t = 5.


Expand Hint
Velocity is the derivative of distance.
Hint 2
The power rule for the first derivative:
$$$\frac{d}{dx}[x^n]=n\cdot x^{n-1}$$$
Velocity is a derivative of distance:
$$$v=\frac{dr}{dt}$$$
where $$v$$ is the instantaneous velocity, $$t$$ is time, and $$r$$ is position.
The power rule for the first derivative:
$$$\frac{d}{dx}[x^n]=n\cdot x^{n-1}$$$
Thus,
$$$D=10t^4-t^3$$$
$$$v=(4)10t^{4-1}-3t^{3-1}=40t^3-3t^2$$$
For $$t=5$$ , substitute "5" in the equation for time:
$$$v=40(5)^3-3(5)^2=5000-75=4,925$$$
4,925


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