Proving the Velocity Formula: vf^2 = vi^2 + 2aΔd

[killaI9BI]

Homework Statement

Can you prove that v_f² = v_i² + 2aΔd?

Homework Equations

The Attempt at a Solution

I don't know where to start. I've not been given any values to use so I'm not sure how to go about answering the question.

 

[mfb]

What did your course cover so far?
This is just an application of energy conservation, but I don't know what you can use in such a proof.

 

[killaI9BI]

This lesson is about acceleration. It's covered vectors and vector components, relative velocity and displacement so far.

 

[matineesuxxx]

This lesson is about acceleration. It's covered vectors and vector components, relative velocity and displacement so far.

If you know any of the other kinematic equations then write them down and think about how you can eliminate time as a variable. are there any combinations or substitutions you can make given any of the other kinematic equations?

 

[killaI9BI]

Maybe it'll help if I post the entire problem:

Derive an equation that relates v_i, v_f, Δd, and a. (Hint: Notice that Δt is not invloved.) Solve for Δt in the first equation. Substitute that value into Δt in the third equation. Solve for v_f². Can you prove that v_f² = v_i² + 2aΔd?

I think that this equation relates all of the variables that the problem is asking for:

v_f = \sqrt[]{}{v_i² + 2a X d}

There is a table on the page that the problem doesn't reference specifically but I'm now realizing that it must be relevant:

solve for Δt for the first equation:

Δt = (v_f - v_i) X a

Substitute that value into Δt for the third equation:
Δd = ((v_i - v_f)/2) X ((v_f - v_i) X a)

Solve for v_f²:
v_f² = v_i² + 2a X d

I'm not sure if I did all of that correctly but it leaves me with the initial problem of proving that that v_f² = v_i² + 2aΔd

I'm not sure I understand where to start with that.

 

[matineesuxxx]

solve for Δt for the first equation:

Δt = (v_f - v_i) X a

Substitute that value into Δt for the third equation:
Δd = ((v_i - v_f)/2) X ((v_f - v_i) X a)

Solve for v_f²:
v_f² = v_i² + 2a X d

I'm not sure if I did all of that correctly but it leaves me with the initial problem of proving that that v_f² = v_i² + 2aΔd

I'm not sure I understand where to start with that.

What do you mean you still have to prove it? you started with true relations and ended up with the required equation; that's how this equation comes about. That's all you need to do.

 

[killaI9BI]

My answer should then be yes I can prove that equation because... ?

Like I said, I'm not sure I understand where to start with that.

 

[matineesuxxx]

what you did is proof enough. In this sense derivation would be proof, because like I said, you are taking relations that are already known to be true, and ending up with the time independent kinematic equation.

 

[killaI9BI]

I really do appreciate your input. I am pretty stunned when it comes to this stuff...

I just don't understand how what I've done above is enough because as you can see here:

Proving that equation is the last problem. I have no idea how to answer it.

 

[Chestermiller]

Rewrite the equation as $v_f^2-v_i^2=(v_f+v_i)(v_f-v_i)=2aΔd$

How is $(v_f-v_i)$ related to a and Δt?

How is $(v_f+v_i)$ related to the average velocity?

How is Δd related to the average velocity and Δt?

Chet

 

[matineesuxxx]

Rewrite the equation as $v_f^2-v_i^2=(v_f+v_i)(v_f-v_i)=2aΔd$...

Could this really be considered a proof? One could easily derive this relation through this method as well.

 

[Chestermiller]

Could this really be considered a proof? One could easily derive this relation through this method as well.

Who knows what they were thinking when they asked for a proof?

Chet

 

[killaI9BI]

lol thanks for your help guys!