Hill Bomb

#412 / Dynamics Math Medium

Consider a skateboarder is bombing down a hill in a straight line so that his distance (D) from a point on the pavement after time (t) is D = 4 t^5 + 3 t. Determine the rider’s acceleration when t = 3.

Expand Hint
Velocity is a derivative of distance.
Hint 2
Acceleration is a derivative of velocity.
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,
$$$D=4t^5+3t$$$
$$$v=(5)4t^{5-1}+(1)3t^{1-1}=20t^4+3t^0=20t^4+3$$$
Acceleration is a derivative of velocity.
$$$a=\frac{dv}{dt}$$$
where $$a$$ is instantaneous acceleration, $$t$$ is time, and $$v$$ is instantaneous velocity.
Applying the power rule for the first derivative to the velocity function:
$$$a=(4)20t^{4-1}+0=80t^3$$$
Finally, substitute $$t=3$$ :
$$$a=80(3)^3=80(27)=2,160$$$
2,160


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