Finding velocity and acceleration components from position vector

[tjbateh]

Homework Statement

A particle moves so that its position (in meters) as a function of time (in seconds) is R= ln(1+t)_i+ (e^t-1)_j + t_k . Write expressions for (a) x-, (b) y- and (c) z-components of its velocity as well as the (d) x-, (e) y- and (f) z-components of its acceleration as functions of time.

Homework Equations

The Attempt at a Solution

x-component= 1/1+t
y-component= e^t
z-component= 1

How do I find their components of acceleration as a function of time??

 

[kuruman]

Take the derivative of a particular position component with respect to time twice. For example

$$v_{x}=\frac{dx}{dt}$$

$$a_{x}=\frac{dv_{x}}{dt}$$

 

[tjbateh]

Ok so for the x-component 1/1+t...the derivative of that would be ...?? I am not good with fractions..Once I have that, I put it over 2t?

 

[kuruman]

Write the fraction as (1 + t)^-1, then take its derivative the usual way.

 

[tjbateh]

so...-(1+t)^-2?

then the next one will just say e^t?
And the final one is 0??

I found that derivative, but what do you mean with the twice time? Or is my work done?

 

[kuruman]

You have found that v_x=-(1+t)^-2. To find the acceleration, you need to take the derivative one more time. So, to get the acceleration, you need to take the derivative two times, one to get the velocity and one more to get the acceleration. Once you get that, your work is not done yet. You need to repeat for the y and z components of the position.

 

[tjbateh]

Ok I put it for the x ..2(1+t)^-3...and it was wrong..That is the second derivative though?

 

[kuruman]

That is the second derivative of x, i.e. the x-component of the acceleration. What exactly do you mean "put it for the x"? What was the question that you used it as an answer to?

 

[tjbateh]

I used that answer for part D, the x-component of acceleration.

 

[kuruman]

I am sorry I misled you. I looked at what you had written down

x-component = 1/(1+t)

and I assumed that this was the x-component of the position. It is actually the velocity component. So you need to take the derivative of that to get the acceleration, which in this case gives you a_x=-(1+t)^-2. I apologize for the confusion it might have caused you. Your work was actually done when you asked if it was.

 

[tjbateh]

No problem! I put the first derivatives in as the answers, and its right! Thanks for the help, and don't worry about the confusion, its ok!