Final answer:
To find the velocity v(t) of an object in free fall at time t, integrate the acceleration due to gravity, which is -9.8 m/s² if up is positive. For an object that starts from rest, the velocity equation simplifies to v(t) = -gt. For example, after 2 seconds, an object's velocity would be -19.6 m/s.
Step-by-step explanation:
The question involves finding the velocity v(t) of an object subject to gravitational acceleration g. This is a classic physics problem under the category of one-dimensional motion involving gravity. If we define the upward direction as positive, then the acceleration due to gravity a(t) would be -9.8 m/s².
To find the velocity v(t) at any time t, we integrate the acceleration function, recognizing that velocity is the integral of acceleration over time. If an object starts from rest, its initial velocity v0 is zero. Therefore, the velocity equation derived from the acceleration due to gravity is given by v(t) = v0 + gt, where v0 is the initial velocity. If the object was dropped, v0 = 0, simplifying our equation to v(t) = -gt.
For instance, if an object is in free fall for 2 seconds, the velocity at that moment would be v(2s) = -9.8 m/s² * 2s = -19.6 m/s, indicating that the object is moving downward with an increasing velocity due to gravity, assuming no air resistance or other forces are acting on it.