Physics - Motion in a Striaght Line - Motion with Constant Acceleration
Let's assume an object moves along the x-axis. The object's acceleration will be Ax (acceleration in the x direction). This makes it simple to derive equations for the position of the object on the x-axis and the Vx (velocity in the x direction)as functions of time.
EQ 1: Ax = (V2x - V1x) / (t2 - t1) or in words, ending velocity minus starting velocity, divided by ending time minus starting time.
Now let t1 = 0 and let t2 be any later time t. We use Vox, or velocity at the initial time t = 0. The x velocity at the later time t2 is Vx.
EQ 1 then becomes:
Ax = (Vx - Vox) / (t - 0) or ending velocity minus initial velocity, divided by ending time minus zero.
This equation can be even further simplified:
EQ 2 : Vx = Vox + (Ax * t) or Final velocity equals initial velocity plus acceleration times final time. This equation is only valid for problems with constant x acceleration.
The term (Ax * t) is the product of the constant rate of change of x-velocity, Ax, and the time interval, t. Therefore it equals the TOTAL change in x-velocity from the initial time (t = 0) to the later time, t. The x velocity, Vx, at any time, t, then equals the initial velocity, Vox at t = 0, plus the change in x velocity, (Ax * t).
Next is to form an equation for the position of the object on the x-axis as a function of time when the acceleration is constant in the x-direction.
Vavx = (X - Xo) / t or The average velocty in the x-direction equals final position minus initial position divided by the time interval.
Another way to derive an average velocity formula is by simply adding the initial and final velocities and dividing by 2. Note: this is ONLY TRUE when acceleration is constant in the x-direction.
EQ 3 : Vavx = (Vox + Vx) / 2 or, the average velocity in the x-direction equals the initial velocity plus the final velocity divided by two.
EQ 2 tells us that Vx can be stated a different way, if we replace that definition into EQ 3, we get,
Vavx = 1/2(Vox + Vox + [Ax * t]) simplified,
Vavx = Vox + 1/2(Ax * t)
From these formulas we can derive our x-position formula:
x = Xo + (Vox * t) + 1/2(Ax * t^2) Valid only for constant x-acceleration.