Final answer:
To find the velocity of a particle, differentiate its position function with respect to time once. To find acceleration, either differentiate the velocity function once or the position function twice with respect to time.
Step-by-step explanation:
Finding Velocity and Acceleration from a Position Function
The velocity of a particle is found by taking the first derivative of the position function with respect to time, while the acceleration is found by taking the second derivative of the position function with respect to time. If the equation of motion (position as a function of time) is provided, then we can differentiate it with respect to time to find these quantities.
For example, given a position function s(t), the velocity v(t) would be ds/dt, and the acceleration a(t) would be the second derivative d²s/dt². From the provided examples, if the position function is r(t) = (50 m/s)tî – (4.9 m/s²)t² Ƶ, the velocity v(t) would be the first derivative of r(t) with respect to time, and the acceleration a(t) is the second derivative of r(t).
In sum, to find velocity as a function of time, differentiate the position function once, and to find acceleration, differentiate the velocity function (or differentiate the position function twice).