A Train's Velocity

#158 / Dynamics Math Easy

At one section of the track, a train travels in a straight line so that its distance (D) from a point on the tracks after time (t) is D = 5t^5 - t^4. Determine the train's velocity when t = 5.

Expand Hint
Velocity is a 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,
$$$D=5t^5-t^4$$$
$$$v=(5)5t^{5-1}-(4)t^{4-1}$$$
$$$v=25t^4-4t^3$$$
Next, substitute $$t=5$$ :
$$$v=25(5)^4-4(5)^3$$$
$$$v=15,625-500=15,125$$$
15,125


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