If v is the velocity, then it is dv/dt.
If we start with v= dx/dt as the instantaneous change in position, then it is d2x/d t2
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if under uniform acceleration or deceleration v = u + (a*t) where: v = instantaneous velocity u = initial velocity a = acceleration (negative if decelerating) t = time elapsed
you are still. motion is at rest.
Suppose you accelerate in your car from stopped to 50 km.hr-1. When you were stopped your instantaneous speed was zero. At the end of the period of acceleration your instantaneous speed was 50 km.hr-1. If your rate of acceleration was constant then your average speed was 25 km.hr-1.
Gravity is a force, which means that it has a corresponding acceleration (rate of rate of change). Because calculus is the study of rates of change, accelerations are studied in calculus.
Measures of motion (displacement, velocity, acceleration) and forces are all vectors so any study involving these would require vector calculus.