Question

A ball is dropped from a boat so that it strikes the surface of a lake with a speed of 16.5 fts. while in the water the ball experiences an acceleration of a = 10 - 0.8v, where a and v are expressed in fus and ft/s, respectively. 1- Find the velocity of the ball as a function of time. 2- If the ball takes 3 s to reach the bottom of the lake, determine the depth of the lake.

Answer 1

To find the velocity function of the ball dropped in water, a differential equation is solved. The depth of the lake is calculated by integrating the velocity function over 3 seconds.

Step-by-step explanation:

To find the velocity of the ball as a function of time when it is dropped into water where its acceleration is given as a = 10 - 0.8v, we first note that acceleration is the derivative of velocity with respect to time (a = dv/dt). We can set up a differential equation and solve for v using separation of variables:

dv / (10 - 0.8v) = dt.

Integrating both sides gives us the velocity as a function of time.

For the second part, to determine the depth of the lake, we'd utilize the velocity function found from the first part and integrate it with respect to time over the given interval of 3 seconds to find the total displacement, which corresponds to the depth of the lake.