I'm trying to derive the formula
v^2 = v0^2 + 2a(x-x0)
were zeros are subscripts
my book tells me to derive it this way
use the definition of average velocity to derive a formula for x
use the formula for average velocity when constant acceleration is assumed to derive a formula for time
rearange the defintion of aceleration for a formula for t
then combine equations to get the derived formula for v^2
so here's my work please show me were I won't wrong
def of average velocity = t^-1 (x - x0)
(average velocity = t^-1(x-x0))t=(avearge velocity)t + x0= x - x0 + x0 = x = (average velocity)t + x0
x = (average velocity)t + x0
def of average velocity were costant acceleration is assumed = 2^-1(v0 + v)
plug into
x = (average velocity)t + x0
x = 2^-1(v0 + v)t + x0
def of acceleration = t^-1(v-v0)
(a=t^-1(v-v0))t=(at=(v-v0))a^-1 = t = a^-1(v-v0)
t = a^-1(v-v0)
plug into x = 2^-1(v0 + v)t + x0
x = 2^-1(v0 + v)a^-1(v-v0) + x0
solve for v^2
x = 2^-1(v0 + v)a^-1(v-v0) + x0
simplfy
x = (a2)^-1(v^2 -v0^2)+ x0
(x = (a2)^-1(v^2 -v0^2)+ x0)2a
(2a)x = (v^2-v0^2) + x0
(2a)x - x0 = (v^2-v0^2) + x0 - x0
(2a)x - x0 + v0^2= (v^2 - v0^2) + v0^2
(2a)x - x0 + v0^2 = v^2
so here's what I got for my equation
v^2 = v0^2 +(2a)x - x0
here's what I was suppose to get
v^2 = v0^2 + 2a(x-x0)
please show me were I went wrong
thank you!
Show me step by step and as to why because I thing your suppose to subtract x0 from both sides why do you multiply???
You want to show that
V^2 - Vo^2 = 2 a (X - Xo)
first rewrite that as:
(V-Vo)(V+Vo) = 2a (X-Xo)
now
your average velocity during the interval = (V+Vo)/2
so
X = Xo + t(V+Vo)/2
but
V = Vo + a t
so
t = (V-Vo)/a
so
X = Xo +(V-Vo)(V+Vo)/2a
we are there.
Let's go through the derivation step by step to identify where you went wrong:
1. Start with the definition of average velocity:
Average velocity = (x - x0) / t, where t is the time interval.
Rearranging the equation: x - x0 = (average velocity) * t
2. Now, use the formula for average velocity when constant acceleration is assumed:
Average velocity = (v0 + v) / 2, where v0 is the initial velocity and v is the final velocity.
Replace average velocity in the previous equation with (v0 + v) / 2:
x - x0 = [(v0 + v) / 2] * t
3. Rearrange the definition of acceleration to get a formula for time:
a = (v - v0) / t, where a is the constant acceleration.
Rearrange the equation to solve for t: t = (v - v0) / a
4. Substitute the expression for time (t) from the previous step into the equation:
x - x0 = [(v0 + v) / 2] * [(v - v0) / a]
5. Simplify the equation by multiplying both sides by 2a:
2a(x - x0) = (v0 + v) * (v - v0)
6. Expand and simplify the right side of the equation using the distributive property:
2a(x - x0) = v*v - v0*v - v0*v + v0*v0
= v^2 - v0^2
7. Finally, isolate v^2 by adding x0 to both sides of the equation:
2a(x - x0) + v0^2 = v^2
So, the correct derived formula is:
v^2 = v0^2 + 2a(x - x0)
In your derivation, it seems like you made a mistake during step 6, where you incorrectly expanded the right side of the equation. Instead of getting v^2 - v0^2, you ended up with (v^2 - v0^2) + 2v0^2.
Remember to be careful with your algebraic manipulations and ensure that you distribute and combine terms correctly.