Mathematics

Problem

A particle moves along a straight line so that its position at time

t

seconds is given by the polynomial function

s(t)=3t^{5}-10t^{3}+4

(position is measured in meters). (a) Calculate the instantaneous velocity

v(t)

of the particle as a function of

t

. Express your answer in meters per second. (b) Evaluate

v(2)

and interpret its physical meaning in the context of the motion. (c) Determine the equation of the line tangent to the graph of

y=s(t)

at the point corresponding to

t=2

. Express your answer in point-slope form.

Intuitive explanation

The key is recognizing that velocity and acceleration are just derivatives of position—they tell you how fast and in what direction the particle is moving at any instant. To understand the particle's motion fully, you'll need to find $s'(t)$ for velocity and $s''(t)$ for acceleration, then use these to identify when the particle speeds up, slows down, or changes direction. Think of the derivatives as tools that reveal the hidden dynamics buried in that polynomial.

Intuitive explanation

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