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Problem 3. A small frictionless cart is attached to the wall by a spring. At ti...

Oct 25, 2024

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Solution by Steps

step 1

To find the cart's maximum speed, we first need to determine the velocity function by differentiating the position function

s(t) = 10 \cos(\pi t)

. The velocity

v(t)

is given by:

v(t) = \frac{ds}{dt} = -10\pi \sin(\pi t)

step 2

The maximum speed occurs when

|\sin(\pi t)| = 1

. This happens at

t = \frac{1}{2} + n

for

n \in \mathbb{Z}

. The maximum speed is

|v(t)| = 10\pi

step 3

The cart is moving the fastest at the positions where

s(t) = 10 \cos(\pi t)

when

\sin(\pi t) = \pm 1

. Thus, the positions are

s\left(\frac{1}{2}\right) = 10 \cos\left(\frac{\pi}{2}\right) = 0

step 4

To determine the acceleration of the cart as a function of time, we differentiate the velocity function:

a(t) = \frac{dv}{dt} = -10\pi^2 \cos(\pi t)

step 5

The magnitude of the acceleration is greatest when

|\cos(\pi t)| = 1

, which occurs at

t = n

for

n \in \mathbb{Z}

. The maximum acceleration is

|a(t)| = 10\pi^2

step 6

The position of the cart when the magnitude of the acceleration is greatest is

s(n) = 10 \cos(0) = 10

cm at

t = 0

and

t = 1

seconds. The speed at this point is

v(0) = 0

and

v(1) = 0

Answer

Maximum speed:

10\pi

cm/s; occurs at

t = \frac{1}{2} + n

seconds. Fastest position:

s = 0

cm. Acceleration function:

a(t) = -10\pi^2 \cos(\pi t)

. Greatest acceleration:

10\pi^2

cm/s² at

s = 10

cm, speed at this point:

0

cm/s.

Key Concept

The motion of a cart attached to a spring can be analyzed using calculus to find speed and acceleration.

Explanation

By differentiating the position function, we can find the velocity and acceleration, allowing us to determine maximum speed and acceleration points.