Can you derive the formula: Vf = Vi +- gt?

Question

Can you derive the formula: Vf = Vi +- gt?

Answer 1

The easiest way is to recognize that acceleration is the rate of change of the velocity.  If we assume constant acceleration, then the acceleration is given by the change in velocity divided by the corresponding change in time.  The change in velocity is the difference between the final velocity and the initial velocity, so we have

a = (vf - vi) / t

where vi is initial velocity, vf is final velocity, and t is the time it took for the object to go from velocity vi to velocity vf.

Rearranging,

vf = vi + at

Now if we are dealing with the acceleration due to gravity, then if we define the positive direction to be the vertically up direction, then the acceleration of gravity is -g where g=9.8m/s^2.

So we have vf = vi - gt.

Answer 2

it seems like the formula is for "final velocity = initial velocity + gravity constant * times" so the derivative will be +- g.

Answer 3

Assuming acceleraiton is constant as g:

a = dv/dt = g

dv = g dt

Integrate the left side from Vi to Vf and the right side from ti to tf

(vf-vi) = g(tf-ti)

Assuming that the initial time is t = 0, tf-ti = tf = t

vf - vi = gt

vf = vi + gt

Of course as the others showed you can derive this from common sense. Acceleration is the change in velocity per unit time. So assuming the acceleration constant, this rate of change is a constant and is just the "slope" of the acceleration graph, so find the change in velocity and divide it by the change in time.

(Vf-Vi)/(t) = a = g

gt = Vf-Vi

Vi+gt = Vf

Same thing

Answer 4

acceleration is defined as the rate of change in velocity. Thus:

dV/dt = +-a

dV = +-adt

V = +-at + C where C is constant of integration

If t = 0, V = the initial velocity Vi. Therefore C = Vi. Thus:

V = Vi +- at

You may denote V as Vf if you regard it as the final velocity. Also you may replace a with g if you are dealing with downward or upward motion where the acceleration is the acceleration of gravity.