Velocity of a particle when acceleration based on displacement

[ivanallen]

Homework Statement

[PLAIN]http://img690.imageshack.us/img690/5817/problem001.jpg

Homework Equations

Chain rule : dv/dt=dv/dr*dr/dt
Integration

The Attempt at a Solution

I can only find the speed (the magnitude of the velocity), that is
[PLAIN]http://img268.imageshack.us/img268/4996/95620213.jpg

I have no idea how to consider its direction.
Does it involve vector integration or higher mathematic knowledge?
Help me, please, I'm dying to know the answer, thank you. :!)

 

[inky]

Where is the problem?

 

[ivanallen]

Problem is I can't find the direction of the velocity of this paricle, the only thing I can find is its magnitude.
In the question above, it asks for the velocity, so only magnitude is not enough.
Please show me how to find its direction. Thanks.

 

[ehild]

You have a vector equation, one equation for each component, x, y, z. ehild

 

[rwisz]

First of all, great job on using the chain rule to find the velocity! You are correct that the magnitude of the velocity can be found by taking the derivative of the displacement with respect to time. However, in order to find the direction of the velocity, we need to take into account the direction of the displacement and the acceleration.

To do this, we can use vector calculus. The velocity can be written as a vector, with components in the x, y, and z directions. We can then use the chain rule to find the components of the velocity vector:

vx = (dx/dt)(dt/dr) = dx/dr * (dr/dt)^-1
vy = (dy/dt)(dt/dr) = dy/dr * (dr/dt)^-1
vz = (dz/dt)(dt/dr) = dz/dr * (dr/dt)^-1

So, in order to find the direction of the velocity, we need to know the direction of the displacement (dx/dr, dy/dr, dz/dr) and the direction of the acceleration (d^2x/dr^2, d^2y/dr^2, d^2z/dr^2). These can be found by taking the second derivative of the displacement with respect to time. Once we have these values, we can use vector addition to find the direction of the velocity.

To summarize, in order to find the velocity of a particle when acceleration is based on displacement, we need to use the chain rule to find the magnitude of the velocity and vector calculus to find the direction of the velocity. This may involve higher level mathematical concepts, but it is important to understand these concepts in order to fully understand the motion of particles in space. Keep up the good work!