*****Please answer all the questions, not just the part a above***** A small fri

Question

*****Please answer all the questions, not just the part a above*****

A small frictionless cart is attached to a wall by a spring. It is pulled 8 cm from its rest position, released at time t = 0. and allowed to roll back and forth for 5 seconds. Its position at time t is s = 8 cos (pi t), a. What is the cart's maximum speed? When is the cart moving that fast? Where is it then? What is the magnitude of the acceleration then? b. Where is the cart when the magnitude of the acceleration is greatest? What is the cart's speed then? a. The maximum speed of the cart is cm/sec. (Round to the nearest tenth as needed.)

Explanation / Answer

s = 8cos(pi*t)

Deriving :

v = ds/dt = -8pi*sin(pi*t) = 0

a = dv/dt = -8*pi^2*cos(pi*t)

for max speed, we have to do dv/dt = 0
-8*pi^2*cos(pi*t) = 0
cos(pi*t) = 0
This happens at t = 1/2 , 3/2 , 5/2 , 7/2 and 9/2

Clearly v = -8pi*sin(pi*t) ---> obtained above

So, this will have a max value when sin(pi*t) = -1(least)
cuz then negative*negative becomes positive

This means,
t = 3/2 and 7/2

And the speed at this time is :
-8pi*sin(pi*t)
-8pi*sin(3pi/2)
-8pi(-1)
8pi

So, max speed of cart = 8pi or 25.1 cm/sec ----> ANSWER

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When is the cart moving that fast?
When t= 3/2 and 7/2

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Where is it then?
s = 8cos(pi*t)
When t = 3/2 or 7/2, we get s = 0
It is at the original position of s = 0 at that point in time

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a = dv/dt = -8*pi^2*cos(pi*t)

Clearly for max speed, acceleration = 0

a = 0

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b)
Acceleration greatest :

dv/dt = -8*pi^2*cos(pi*t)

Deriving this,
we get
a' = 8pi^3 * sin(pi*t) = 0
sin(pi*t) = 0
t = 0,1,2,3,4,5 works

Now, acc, dv/dt = -8*pi^2*cos(pi*t)
would be greatest when cos(pi*t) = -1
And this happens when t = 1 , 3 , 5

When t = 1, s= 8cos(pi)= -8
When t = 3,s = 8cos(3pi) = -8 again
When t = 5, s = -8 again

So, the cart is at position s = -8 when the acceleration is greatest

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Carts speed then :
We need to find the speed when t = 1,3,5...

v = ds/dt = -8pi*sin(pi*t)

Clearly since we have sin(pi*t) there,
and since sin(pi) , sin(3pi) and sin(5pi) are all 0,
v = 0 at this time

v = 0

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