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!
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
OK so far
(x = (a2)^-1(v^2 -v0^2)+ x0)2a
(2a)x = (v^2-v0^2) + x0 the last term should be xo*2a ng
Your approach started off correctly, but there are a few errors in your derivation.
Let's go through it step by step:
Step 1: Use the definition of average velocity to derive a formula for x.
You correctly wrote the equation:
x = (average velocity)t + x0
Step 2: Use the formula for average velocity when constant acceleration is assumed to derive a formula for time.
The correct formula for average velocity when constant acceleration is assumed is:
average velocity = (v + v0) / 2
To derive a formula for time, we rearrange the equation in terms of time:
x = (average velocity)t + x0
2 * average velocity * t = v + v0
2 * t = (v + v0) / average velocity
t = (v + v0) / (2 * average velocity)
Note: Make sure to use parentheses correctly to avoid incorrect calculations later.
Step 3: Rearrange the definition of acceleration for a formula for time.
The correct formula for acceleration is:
acceleration = (v - v0) / t
To rearrange the equation for time, we have:
acceleration = (v - v0) / t
t = (v - v0) / acceleration
Step 4: Combine equations to get the derived formula for v^2.
Now, we substitute the expressions for time from Step 2 and Step 3 into the equation:
x = (average velocity)t + x0
x = (average velocity) * (v + v0) / (2 * average velocity) + x0
x = (v + v0) / 2 + x0
We simplify this equation to:
(v + v0) / 2 = x - x0
(v + v0) = 2(x - x0)
v = 2(x - x0) / (v0 + v) -- Equation 1
Next, substitute the expression for time from Step 3 into the equation:
t = (v - v0) / acceleration
t = (v - v0) / a
Rewrite t as ((x - x0) * 2) / (v - v0) from Equation 1:
((x - x0) * 2) / (v - v0) = (v - v0) / a
Cross-multiply and simplify:
2(x - x0) = a(v - v0)
2x - 2x0 = av - av0
av = 2x - 2x0 + av0
av = 2(x - x0) + av0
v = 2(x - x0) / a + v0
To derive the formula for v^2, square both sides of the equation:
v^2 = (2(x - x0) / a + v0)^2
v^2 = (4(x - x0)^2 / a^2) + (8(x - x0)v0 / a) + v0^2
v^2 = (4(x - x0)^2 / a^2) + (8(x - x0)v0 / a) + v0^2
Now, let's rearrange the equation to match the intended formula:
v^2 = v0^2 + 4(x - x0)^2 / a^2 + 8(x - x0)v0 / a
Simplifying further:
v^2 = v0^2 + 4(x - x0)^2 / a^2 + (8v0(x - x0)) / a
Finally, let's write it in the desired form:
v^2 = v0^2 + 2(x - x0) / a + (4v0(x - x0)) / a
Thus, the correct derived formula for v^2 is:
v^2 = v0^2 + 2(x - x0) * a + 4v0(x - x0)
Let's go through your derivation step-by-step to identify where the mistake occurred.
1. You correctly started with the definition of the average velocity as t^(-1) (x - x0).
2. Then you wrote x = (average velocity) * t + x0, which is correct.
3. Next, you used the formula for average velocity when constant acceleration is assumed, v = 2^(-1) (v0 + v). However, you made a mistake in substituting this into the previous equation. It should be x = (average velocity) * t + x0 = 2^(-1) (v0 + v) * t + x0 (not 2^(-1) (v0 + v) * t + x0).
4. You correctly wrote the definition of acceleration as a = t^(-1) (v - v0).
5. Then you made a mistake when substituting t = a^(-1) (v - v0) into x = 2^(-1) (v0 + v) * t + x0. It should be x = 2^(-1) (v0 + v) * t + x0 = 2^(-1) (v0 + v) * (a^(-1) (v - v0)) + x0 (not 2^(-1) (v0 + v) * t + x0).
6. When simplifying the expression, you made an error. It should be x = 2^(-1) (v0 + v)(a^(-1))(v - v0) + x0.
7. You mistakenly wrote (x = (a2)^(-1)(v^2 - v0^2) + x0)2a instead of simplifying the expression as (2a)x = (v^2 - v0^2) + x0.
8. Continuing with the simplification, you incorrectly wrote (2a)x - x0 + v0^2 = (v^2 - v0^2) + v0^2. It should be (2a)x - x0 + v0^2 = (v^2 - v0^2) - v0^2.
9. Finally, you made an error in combining the terms, resulting in v^2 = v0^2 + (2a)x - x0 + v0^2, instead of v^2 = v0^2 + 2a(x - x0).
To correct the mistake, you need to fix step 5 and continue the simplification correctly. Here's the corrected derivation:
1. Definition of average velocity: v = t^(-1) (x - x0).
2. Rearrange to solve for x: x = vt + x0.
3. Formula for average velocity with constant acceleration: v = 2^(-1) (v0 + v).
4. Substituting into x = vt + x0: x = 2^(-1) (v0 + v) t + x0.
5. Definition of acceleration: a = t^(-1) (v - v0).
6. Substituting t = a^(-1) (v - v0) into x = 2^(-1) (v0 + v) t + x0:
x = 2^(-1) (v0 + v) (a^(-1) (v - v0)) + x0.
7. Simplifying the expression: x = 2^(-1) (v0 + v) (a^(-1)) (v - v0) + x0.
8. Multiply through by 2a: 2ax = v0v - v0^2 + v^2 - v0v + 2x0a.
9. Combine like terms: 2ax - 2x0a = v^2 - v0^2.
10. Rearrange the equation: v^2 = v0^2 + 2a(x - x0).
So the correct derivation is:
v^2 = v0^2 + 2a(x - x0)