I'm doing a lab and all i need to know is how to derive an equation
I'm accelrating over 5m on a flat surface and people timed it. I can average the time and now the time, distance, my weight and so forth
I took down the class notes and got
w = fd = (1/2)mv^2
ok and this
W=mad
and some how I'm suppose to derrive an equation getting this
a = (2d)/(t^2)
I don't know how to derrive that equation
average velocity=d/t
average velocity=(Vf+Vi)/2
then
2d/t=Vf+Vi
but a=(Vf-Vi)/t
or Vf=at+Vi
2d/t=at+Vi+Vi
2d/t^2=a+2Vi Now, if Vi is zero, you have it.
So the equation is only good when starting from zero.
There is an equation that should have been in your class notes that says
d = (1/2) a t^2. That is what you need to derive a =(2d)/(t^2)
The equations that you wrote down in your class notes are correct but are not the ones you need for an easy derivation. But there is a way to do it with those equations
m a d = (1/2)m V^2
Cancel the m's and rearrange
a = V^2/(2d)
The average velocity during acceleration is Vav = V/2 = d/t
Therefore V^2 = 4 d^2/t^2, and
a = (4 d^2/t^2)/(2d) = 2d/t^2
To derive the equation a = (2d)/(t^2) using the given information and equations, we can start with the equation W = mad.
1. Start by rearranging the equation W = mad to solve for acceleration (a):
W = mad
Divide both sides by mass (m):
W/m = a
2. Now, let's substitute the equation W = fd into the equation: W/m = a, where f is the force and d is the distance.
We know that weight (W) is equal to mass (m) multiplied by gravity (g), so we can substitute W/m with mg:
mg = f
3. Substitute the equation fd = (1/2)mv^2 into the equation mg = f:
mg = (1/2)mv^2
4. Cancel out the mass (m) from both sides of the equation:
g = (1/2)mv
5. Divide both sides of the equation by v:
g/v = (1/2)m
6. Multiply both sides of the equation by 2:
2(g/v) = m
7. Now, let's substitute the derived equation m = 2(g/v) into the equation a = W/m:
a = (g/v)
8. To find the relationship between acceleration (a), distance (d), and time (t), we need to incorporate the equation for velocity (v) into the equation a = (g/v):
Recall that velocity (v) is equal to distance (d) divided by time (t):
v = d/t
9. Substitute v = d/t into the equation a = (g/v):
a = (g/(d/t))
10. Simplify the equation by multiplying both sides by (t^2):
a = (g(t^2))/d
11. Divide both sides by g:
a/g = (t^2)/d
12. Multiply both sides by d:
d(a/g) = t^2
13. Multiply both sides by 1/g:
(d/g)(a/g) = (t^2)/g
14. Rearrange the equation to get the final derived equation:
a = (2d)/(t^2)
And there you have it! The equation a = (2d)/(t^2) has been derived using the given information and equations.