VERTICAL MOTION

The acceleration due to gravity is a constant near the surface of the Earth. Its accepted values are $-32 \text{ ft}/\text{s}^2$ or $–9.8 \text{ m}/\text{s}^2$. These values are all the information needed to derive the equations of $v(t)$ 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. Write an equation for velocity, $v(t)$, if $a(t) =-32 \text{ ft}/\text{s}^2$ 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. Write an equation for 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):

   $s\left(t\right) = −16t^{2} + 120t + 100$

4. What is the velocity of the object when it hits the ground

   Steps (d):

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

5. What is 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 $s\left(t\right)$ the steepest? Remember that speed is the absolute value of velocity.