Question
Use Eqs. (2.17) where vx = vox + axt and (2.18) where x = xo + ∫vxdt to find vx, and x as functions of time in the case in which the acceleration is constant.
Solution
To address this problem using the method and notation provided in Eqs. (2.17) and (2.18) under constant acceleration:
Eq. (2.17): The velocity as a function of time: vx = vox + axt
Eq. (2.18): The position as a function of time: x = xo + ∫vxdt
Where:
vx = Velocity at time t,
vox = Initial velocity,
ax = Constant acceleration,
xo = Initial position.
1. Find Velocity (vx(t)):
From equation (2.17): vx(t) = vox + axt
Since ax is constant, this simplifies to: vx(t) = vox + axt
2. Find Position (x(t)):
From Eq. (2.18): x(t) = xo + ∫vx(t)dt
Substitute vx(t) = vox + axt into the above equation and you will have:
x(t) = xo + ∫(vox + axt)dt = xo + voxt + (1/2)axt2
The above results are consistent with the standard equations of motion for constant acceleration. Although these equations were developed for cases where ax is constant, they are derived from the general integral forms of Eqs. (2.17) and (2.18), which apply even when ax varies with time.