Astudent throws a set of keys vertically upward to his fraternity brother, who is in a window a distance h above. the brother's outstretched hands catches the keys on their way up at a time t later. (a) with what initial velocity were the keys thrown? (b) what was the velocity of the keys just before they were caught? (answer should be in terms of h, t, and g.)

1) The initial velocity equals

2) The velocity with which the keys were caught equals

Explanation:

Let the keys be thrown with a speed 'u' vertically upwards. Since the keys are caught after time 't' while covering a distance 'h' we can use second equation of kinematics to relate these terms.

According to second equation of kinematics the displacement of an object in 't' time is given by

Applying the corresponding values and solving for the initial speed 'u' we get

The change in velocities while covering a distance 's' can be found from first equation of kinematics

Thus we have

Applying corresponding values we get

we have been given a quadratic function and we need to restrict the domain such that it becomes a one to one function.

we know that vertex of this quadratic function occurs at (5,2).

further, we know that range of this function is .

if we restrict the domain of this function to either or , it will become one to one function.

let us know find its inverse.

upon interchanging x and y, we get:

let us now solve this function for y.

hence, the inverse function would be if we restrict the domain of original function to and the inverse function would be if we restrict the domain to .