using this information: if an object is thrown straight up into the air from the height H feet per second then at time tseconds the height of the object is

-16.1t^2 +Vt+ H feet.
This formula uses only gravitational force, ignoring air friction. It is valid only until the object hits the ground or some other object.

Find a function b such that b(v) is the length of time in seconds that a ball takes to reach its maximum height when thrown straight up with initial velocity V feet per second

To find the function b(v), which gives the length of time in seconds that a ball takes to reach its maximum height when thrown straight up with initial velocity V feet per second, we need to determine at which time the ball reaches its maximum height.

From the given information, we have the formula for the height of the object at time t seconds:

h(t) = -16.1t^2 + Vt + H

When the object reaches its maximum height, the velocity is zero, indicating that the object is momentarily at rest. So, we need to find the time when the velocity of the object is zero.

To determine the time when the velocity is zero, we differentiate the height function with respect to time:

h'(t) = -32.2t + V

Setting h'(t) equal to zero, we get:

-32.2t + V = 0
t = V / 32.2

So, the time it takes for the ball to reach its maximum height is t = V / 32.2 seconds.

Therefore, the function b(v) is given by b(v) = v / 32.2, where v is the initial velocity of the ball in feet per second.

b = (Vf-Vo)/g.

Vf = 0 at max ht.
b = (0-Vo)/32.