Deriving the Formula for Final Velocity in One Dimension

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  • #1
SaltyBriefs
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Homework Statement


I'm trying to derive this formula, but I get stuck after I factor the t out.

V[itex]_{f}[/itex][itex]^{2}[/itex] = V[itex]_{0}[/itex][itex]^{2}[/itex] + 2a (y-y[itex]_{0}[/itex])

Homework Equations



V[itex]_{f}[/itex][itex]^{2}[/itex] = V[itex]_{0}[/itex][itex]^{2}[/itex] + 2a (y-y[itex]_{0}[/itex])

The Attempt at a Solution


1) y[itex]_{f}[/itex] - y[itex]_{0}[/itex] = ([itex]\frac{V_{0}+V_{f}}{2}[/itex])t

2) y[itex]_{f}[/itex] - y[itex]_{0}[/itex] ([itex]\frac{1}{t}[/itex])= ([itex]\frac{V_{0}+V_{f}}{2}[/itex])t ([itex]\frac{1}{t}[/itex])

3) V[itex]_{f}[/itex]= ([itex]\frac{V_{0}+V_{f}}{2}[/itex])t ([itex]\frac{1}{t}[/itex])

4) ?

5) V[itex]_{y}[/itex] = V[itex]_{0y}[/itex][itex]^{2}[/itex] + 2a (y[itex]_{f}[/itex]-y[itex]_{0}[/itex])
 
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  • #2
SaltyBriefs said:

Homework Statement


I'm trying to derive this formula, but I get stuck after I factor the t out.

V[itex]_{f}[/itex][itex]^{2}[/itex] = V[itex]_{0}[/itex][itex]^{2}[/itex] + 2a (y-y[itex]_{0}[/itex])

Homework Equations



V[itex]_{f}[/itex][itex]^{2}[/itex] = V[itex]_{0}[/itex][itex]^{2}[/itex] + 2a (y-y[itex]_{0}[/itex])

The Attempt at a Solution


1) y[itex]_{f}[/itex] - y[itex]_{0}[/itex] = ([itex]\frac{V_{0}+V_{f}}{2}[/itex])t

2) y[itex]_{f}[/itex] - y[itex]_{0}[/itex] ([itex]\frac{1}{t}[/itex])= ([itex]\frac{V_{0}+V_{f}}{2}[/itex])t ([itex]\frac{1}{t}[/itex])

3) V[itex]_{f}[/itex]= ([itex]\frac{V_{0}+V_{f}}{2}[/itex])t ([itex]\frac{1}{t}[/itex])

4) ?

5) V[itex]_{y}[/itex] = V[itex]_{0y}[/itex][itex]^{2}[/itex] + 2a (y[itex]_{f}[/itex]-y[itex]_{0}[/itex])

Firstly, I can only show this using symbols
v for your Vf - final velocity
u for your Vo - initial velocity
s for our y - yo - displacement.
{partially from familiarity, and partly because it is easier to type}

So I am aiming at v2 = u2 + 2as

you are hopefully familiar with a couple of other motion equations

v = u +at & s = t*(v + u)/2 [you listed tis second one in line 1 of your solution]

These two are combined.

The first can be transposed to give

t = (v - u)/a

substitute for t in the second

s = (v-u)(v+u)/2a

so

(v-u)(v+u) = 2as

v2 - u2 = 2as

or

v2 = u2 + 2as
 
  • #3
Alternatively:

a = dv/dt

a(ds) = ds/dt(dv)

int(ads) = int(vdv)

Assuming a = constant.

as = 1/2(v^2-u^2)

v^2 = u^2 + 2as
 
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