Final answer:
The change in the ball's horizontal position is found by integrating the velocity function between times 0 and 3.8 seconds. The acceleration at t = 0 is 0 m/s², while the acceleration at t = 3.8 seconds is calculated using the derivative of the velocity function, giving 9(3.8)² m/s².
Step-by-step explanation:
The velocity of a ball in the horizontal direction is given by the function v(t) = 20 + 3t³, where t is in seconds and the velocity is in meters per second.
a) Change in Position
To find the change in position between t = 0 and t = 3.8 seconds, we need to integrate the velocity function:
∫ v(t) dt = ∫ (20 + 3t³) dt
Calculating the integral from 0 to 3.8 gives the change in position, which can be found using basic integration techniques.
b) Acceleration at t = 0
The acceleration is the derivative of the velocity function with respect to time:
a(t) = d/dt (20 + 3t³) = 9t²
At t = 0, the acceleration a(0) = 0 m/s².
c) Acceleration at t = 3.8 s
Using the same formula for acceleration, at t = 3.8 seconds:
a(3.8) = 9(3.8)² m/s²
This value can be calculated to find the acceleration at that specific point in time.