The Equation of Motion x(t)
The notation x(t) is a shorthand way of symbolizing the fact that an object's location "x" can be expressed as a explicit function of time, x = f(t).
Examples:
For an object undergoing constant acceleration, f(t) = 1/2 a t2 + vo t + xo
For an object executing simple harmonic motion, f(t) = A sin(w t + f)
Note that in addition to being a function of time, x(t) is also a function of other variables. We do not usually express x's explicit dependence upon these other variables, but it is implied.
- Once x(t) is explicitly known, then the history of the motion of the object can be determined.
Location: xA= x(tA)
You know where the object is located at any moment tA by evaluating x(t) at t = tA.
Velocity: vA= v(tA) where v(t) = dx/dt
You know how fast the object is moving at any moment tA by taking the derivative of x(t) with respect to time to find v(t) and then evaluating v(t) at t = tA.
Acceleration: aA = a(tA) where a(t) = dv/dt
You know the acceleration of the object at any moment tA by taking the derivative of v(t) with respect to time to find a(t) and then evaluating a(t) at t = tA