Final answer:
To find the velocity of the ball after 3 seconds, substitute t = 3 into the position function s(t) = -16t^2 + v0t + s0. To find the velocity after falling 116 feet, solve the equation -16t^2 + (-26)t + 495 = -116.
Step-by-step explanation:
To find the velocity of the ball after 3 seconds, we can substitute t = 3 into the position function, s(t) = -16t² + v₀t + s₀. Since the ball is thrown straight down, the initial velocity v₀ = -26 ft/s and the initial position s₀ = 495 ft. So, we have s(3) = -16(3)² + (-26)(3) + 495. Evaluating this expression gives us s(3) = 237 ft. Therefore, the velocity after 3 seconds is 237 ft/s.
To find the velocity after falling 116 feet, we can set the position function equal to -116 and solve for t: -16t² + (-26)t + 495 = -116. This is a quadratic equation, which we can solve to find the value(s) of t. Substituting the positive root of t into the position function will give us the velocity at that time. Alternatively, we can differentiate the position function with respect to time to obtain the velocity function v(t) = -32t - 26, and then substitute t = the time it took to fall 116 feet to find the velocity.