Final answer:
The average velocity can be found using the change in position over the change in time for the given intervals, and the instantaneous velocity at t=2 is calculated by taking the derivative of the position function and evaluating it at t=2, which gives -24 ft/s.
Step-by-step explanation:
The question at hand involves finding both the average and instantaneous velocities of a ball thrown into the air, described by the function y = 40t - 16t^2, where 'y' represents the height in feet and 't' represents the time in seconds after the ball is thrown.
Average Velocity
The average velocity over a time interval can be found by the formula:
- Average velocity = (Change in position) / (Change in time)
Using this formula, we can find the average velocity for the various time intervals given by plugging in the values of 't' into the equation for 'y' to obtain the position at the start and end of each interval.
Instantaneous Velocity
Instantaneous velocity is the velocity at a specific moment in time and can be estimated by taking the derivative of the position function and evaluating it at the given time. When t=2:
- Instantaneous velocity = dy/dt = 40 - 32t
- Plug in t=2: Instantaneous velocity = 40 - 32(2) = -24 ft/s
The negative value indicates the ball is moving downward at that instant.