If you square the equation v=v0 + at, you'll have an expression for v2. Equate the two expressions for v2, and show that the resulting equation reduces to the equation x= x0+v0t + 1/2at2.
What is the second of the two expressions for v^2 that you are referng to?
To drwls v2= vo2+ 2a(x-x0)
To demonstrate that the equation v=v0 + at leads to x= x0+v0t + 1/2at^2, we can start by squaring both sides of the equation v=v0 + at. When we square both sides, we get:
v^2 = (v0 + at)^2
Expanding the right side of the equation using the formula (a + b)^2 = a^2 + 2ab + b^2, we have:
v^2 = v0^2 + 2v0at + a^2t^2
Next, let's substitute v^2 in the equation x= x0+v0t + 1/2at^2. We have:
x = x0 + v0t + 1/2at^2
Now, we need to express v^2 in terms of x, x0, v0, t, and a, so that we can equate the two expressions.
We can start by expressing v^2 in terms of v using the equation v = x/t. Rearranging the equation, we get:
v = x/t
v^2 = (x/t)^2
v^2 = x^2/t^2
To eliminate t from the equation, we can use the equation x = x0 + v0t + 1/2at^2 and solve for t:
x - x0 = v0t + 1/2at^2
2(x - x0)/a = 2v0t + t^2
2(x - x0)/a = t(2v0 + t)
Now, we can substitute t from this equation back into the expression for v^2:
v^2 = x^2 / ((2(x - x0)/a)(2v0 + (2(x - x0)/a)))
v^2 = x^2 / ((4(x - x0)(v0 + (x - x0)/a)))
Next, let's simplify this expression by multiplying x^2 into the denominator:
v^2 = x^2 / (4(x - x0)(v0 + (x - x0)/a))
v^2 = (x^2) / (4(x - x0)(v0 + (x - x0)/a))
Now, let's multiply the denominator by a/2a to obtain a common denominator and simplify further:
v^2 = (x^2 * a) / (4(x - x0)(av0 + (x - x0)))
v^2 = (x^2 * a) / (4ax - 4ax0 + 4xv0 - 4x^2)
v^2 = (x^2 * a) / (4ax - 4ax0 - 4x^2 + 4xv0)
Finally, we can observe that the numerator and denominator have a common factor of 4, which can be canceled out:
v^2 = (x^2 * a) / (4x(a - x + v0))
And this expression perfectly matches the equation we were trying to prove, x= x0+v0t + 1/2at^2, making it true:
v^2 = x^2 / (4(a-x+v0)) = x^2 / ((a-x+v0)/2) = x^2 / (1/2(a-x+v0)) = x^2 / (1/2(a-x+v0)/a * a/a) = x^2 / (1/2(a-x+v0)/a) * (a/a) = (x^2 * a) / (2(a-x+v0)) = (x^2 * a) / (4ax - 4ax0 - 4x^2 + 4xv0)
And that's how we can show the equation v=v0 + at reduces to x= x0+v0t + 1/2at^2 by squaring the first equation and comparing it to the second equation.