VERTICAL MOTION

The acceleration due to gravity is a constant near the surface of the Earth. Its accepted values are $−32.2$ ft/sec^{$^{2}$} or $−9.8$ m/sec^{$^{2}$} . These values are all the information needed to derive the equations $v(t)$ and $a(t)$ for vertical motion for any dropped or thrown object.

1. Describe how can you find a velocity function when given the acceleration function.

   Hint (a):

   Acceleration is the rate of change of velocity...

2. Find $v(t)$ if $a(t) = −32$ ft/sec^{$^{2}$} and if the initial velocity of an object, $v_0$, is $120$ feet per second.

   Hint (b):

   There is a relationship between the $+C$ and the initial velocity.

3. Find its height $s(t)$ if the starting position of the object, $x_0$, is $100$ feet above ground.

   Hint (c):

   Velocity is the rate of change of position...

   Answer (c):

   $s(t) = −16t^2 + 120t + 100$

4. Find the velocity of the object when it hits the ground.

   Steps (d):

   First find the TIME when the object hits the ground: $s(t) = 0$.Then evaluate $v(t)$ at that time.

5. Find the maximum height attained by the object.

   Hint (e):

   Find the vertex of the position function. You could use Algebra or Calculus.

6. At what time is the object's speed greatest?

   Hint (f):

   Where is the slope of $s(t)$ the steepest? Remember that speed is the absolute value of velocity.