Deriving Acceleration from Potential Energy?

[cj]

I have a known potential energy, V, expression:

V(x,y,z) = α·x + β·y² + γ·z³

I'm given: @(0,0,0), v = v₀ and then asked to find v at (1,1,1).

I can determine v from Conservation of Energy:

v² = v₀² - (2/m)·(α + β + γ)²

In general, what is the expression for the accelerations a_x, a_y, a_z?

Do I find F from -∇V?

If so, what's next (as far as finding the acceleration's x, y and z-components)?

Thanks!

 

[arildno]

1.Why have you squared the potential energy term??

2. Yes, and divide F by m to find the accelerations.

 

[ZapperZ]

I have a known potential energy, V, expression:

V(x,y,z) = α·x + β·y² + γ·z³

I'm given: @(0,0,0), v = v₀ and then asked to find v at (1,1,1).

I can determine v from Conservation of Energy:

v² = v₀² - (2/m)·(α + β + γ)²

In general, what is the expression for the accelerations a_x, a_y, a_z?

Do I find F from -∇V?

If so, what's next (as far as finding the acceleration's x, y and z-components)?

Thanks!

arildno sort of told you how to start it off. You should have F (and a) from the gradient of V. However, you will notice that "a" has a dependence on x, y, and z. If a is a function of t, then it is trivial to find v. But you don't have that here.

So what you need to do to find v is to use some calculus gymnastics by invoking the chain rule, i.e.

a = dv/dt = (dv_x/dx * dx/dt)i^ + (dv_y/dy * dy/dt)j^ + (dv_z/dz * dz/dt)k^

It is easier to solve this component by component, so for the x-component, you have

a_x = dv_x/dx * v_x (since dx/dt = v_x)

Thus, a_x dx = v_x dv_x

I think you should be able to handle the baby integral here using the initial conditions given. Do the same thing for the other 2 components.

Zz.

 

[cj]

Thanks a lot Zz.

When solving, for example, a_z, I arrive at:

a_z = -(3γ/m)·z²

In general, is this a sufficient expression for a_z,
or should it be reduced or otherwise expressed differently?

arildno sort of told you how to start it off. You should have F (and a) from the gradient of V. However, you will notice that "a" has a dependence on x, y, and z. If a is a function of t, then it is trivial to find v. But you don't have that here.

So what you need to do to find v is to use some calculus gymnastics by invoking the chain rule, i.e.

a = dv/dt = (dv_x/dx * dx/dt)i^ + (dv_y/dy * dy/dt)j^ + (dv_z/dz * dz/dt)k^

It is easier to solve this component by component, so for the x-component, you have

a_x = dv_x/dx * v_x (since dx/dt = v_x)

Thus, a_x dx = v_x dv_x

I think you should be able to handle the baby integral here using the initial conditions given. Do the same thing for the other 2 components.

Zz.

 

[ZapperZ]

Thanks a lot Zz.

When solving, for example, a_z, I arrive at:

a_z = -(3γ/m)·z²

In general, is this a sufficient expression for a_z,
or should it be reduced or otherwise expressed differently?

ASSUMING you did the gradient correctly, that should be a sufficient expression for the a_z to play with.

Zz.