Expert: Paul Klarreich - 11/26/2011

Question
A small frictionless cart, attached to the wall by a spring is pulled 10 cm from its rest position and released at time t=0 to roll back and forth for 4 seconds. its position at time t is s(t)=10cos(pi t)

a. when does the cart reach maximum speed?
b. what is the cart's maximum speed?
c. where is the cart when it has maximum speed?
d. what is the magnitude of the acceleration then?
e. where is the cart when the magnitude of the acceleration is greatest?
f. what is the cart's speed when the acceleration is greatest?

Answer
Questioner: Joshua
Country: California, United States
Category: Calculus
Private: No
Subject: Optimization Problem
Question: A small frictionless cart, attached to the wall by a spring is pulled 10 cm from its rest position and released at time t=0 to roll back and forth for 4 seconds. its position at time t is s(t)=10cos(pi t)

If [the position]:  s(t) = 10 cos(pi t), then:

[the velocity]:     v(t) = - 10 pi sin(pi t),  and:

[the acceleration]:  a(t) = - 10 pi^2 cos(pi t)

Use these principles:

The max or min of a value occurs when ITS derivative is zero:

Max-min position when v = 0
Max-min velocity when a = 0

and:
'WHEN does' always means 'find t such that...'.
'WHERE IS' always means 'find s ...'.

a. when does the cart reach maximum speed?

Max speed means a(t) = 0 :  cos(pi t) = 0  when  pi t = pi/2 or 3pi/2,

so t = 1/2 or 3/2

b. what is the cart's maximum speed?

Find v(1/2)  or v(3/2)

c. where is the cart when it has maximum speed?

Find s(1/2) or s(3/2)

d. what is the magnitude of the acceleration then?  

Didn't we answer this already?  See (a.)

e. where is the cart when the magnitude of the acceleration is greatest?

This means:  find s(t), such that  a'(t) = 0.  

f. what is the cart's speed when the acceleration is greatest?

This means:  find v(t), such that  a'(t) = 0.  

You should be able to finish up now.