Expert: Paul Klarreich - 11/17/2008

Question
I know your instructions said tell me what you've already tried and don't just dump the question on me or ill assume you only need to get started, I do need to know how to get started as well as needing help with the rest of the problem this is my first time doing a problem like this at all...

the position (distance) of a particle moving along a horizontal line is given by: s(t) = t^3-6t^2 for t>or=0  (t in seconds; s in meters)
a) find the velocity function v(t)
b) find the acceleration function a(t)
c) graph s(t), v(t), and a(t) on same axes (i'm assuming this can be done on a calculator very easily)
d) find average velocity from t=1 to t=4
e) on what time interval is the particle moving left?
f) find velocity when a(t) = 0

Answer
Questioner:   Lauren
Category:  Calculus
Private:  No
 
Subject:  help!!
Question:  I know your instructions said tell me what you've already tried and don't just dump the question on me or ill assume you only need to get started, I do need to know how to get started as well as needing help with the rest of the problem this is my first time doing a problem like this at all...

the position (distance) of a particle moving along a horizontal line is given by: s(t) = t^3-6t^2 for t>or=0  (t in seconds; s in meters)
a) find the velocity function v(t)
b) find the acceleration function a(t)
c) graph s(t), v(t), and a(t) on same axes (i'm assuming this can be done on a calculator very easily)
d) find average velocity from t=1 to t=4
e) on what time interval is the particle moving left?
f) find velocity when a(t) = 0
..................................
Hi, Lauren,

A famous mathematician said once that carefully rewording the problem is equivalent to solving it.  So in each of these, all you have to do is reword them:

a) find the velocity function v(t)

Since the velocity is the derivative of the position, this says:

find the derivative  s'(t)


...................
b) find the acceleration function a(t)

Since acceleration is the derivative of the velocity, this says:

find the derivative of v(t), or   v'(t), or s''(t)

...................
c) graph s(t), v(t), and a(t) on same axes (i'm assuming this can be done on a calculator very easily)

yes, it can.


...................
d) find average velocity from t=1 to t=4

Since the average value of any function f(t) over an interval [a,b], is given by
f(b) - f(a)
-----------
b - a
this says:

Find the average value of the velocity function on [1,4].

................
e) on what time interval is the particle moving left?

Since 'moving left' means negative velocity, this means:

Find the values of t that give  v(t) < 0.  So you will write the inequality:

v(t) < 0

and solve it for an interval. [Forget how to solve inequalities?  Let me know.]


.....................

f) find velocity when a(t) = 0

This practically solves itself.  It says:

solve  a(t) = 0
substitute that value of t into  v(t)

NOW YOU CAN GET STARTED.