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### Moving Particle

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Date: 07/14/99 at 13:39:03
From: Kirsten Peterson
Subject: Moving Particle Equation

In my College Calculus class we were given the following problem. I
think I understand the logic behind it but the answers just don't seem

Assume a particle moves on the x-axis according to the formula

x = t^3-6t^2+9t+5

Find:
a. the velocity when t = 3
b. the acceleration when t = 4
c. the times when the velocity is zero
d. the place where the acceleration is zero
e. the acceleration at each of the two moments where the
particle is motionless

This is what I did:

First I differentiated the equation

dx/dt = 3t^2-12t+9

a. velocity when t = 3

I then plugged in 3 for t

dx/dt = 3(3)^2-12(3)+9 = 0

b. I do not remember how to find the acceleration of a particle.
c. logically, with how I did a, the answer to this problem would be
the given in a, because the answer for the velocity is 0 so the
answer to this equation is going to be t = 3. This is where my
logic doesn't make any sense. How can that be the right answer?

Thank you!

Kirsten
```

```
Date: 07/14/99 at 16:32:56
From: Doctor Anthony
Subject: Re: Moving Particle Equation

>Assume a particle moves on the x-axis according to the formula
>     x=t^3-6t^2+9t+5
>Find:
>a. the velocity when t=3

dx/dt = 3t^2 - 12t + 9    and putting t = 3

dx/dt = 27 - 36 + 9 = 0   so the particle is stationary at t = 3.

>b. the acceleration when t = 4

accel = d^2(x)/dt^2 = 6t - 12   and putting t = 4

accel = 24 - 12 =  12 m/sec^2

>c. the times when the velocity is zero.

We solve  3t^2 - 12t + 9 = 0

t^2 - 4t + 3 = 0

(t-1)(t-3) = 0

So the particle is stationary at t = 1 and t = 3.

>d. the place where the acceleration is zero.

6t - 12 = 0

so   accel = 0  when t = 2

and then x = t^3 - 6t^2 + 9t + 5
= 8 - 24 + 18 + 5
= 7 metres

>e. the acceleration at each of the two moments where the particle is
>motionless.

When t = 1   accel = 6t - 12 = -6 m/sec^2

when t = 3   accel = 16 - 12 = +6 m/sec^2

- Doctor Anthony, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
College Calculus
High School Calculus

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