Applications of the Quadratic Formula Solutions

• There are many instances in the real world of objects following a    parabolic    path: fireworks, fountains, throwing or dropping an object.

• The general equation for projectile (parabolic) motion path is: $h=\frac{1}{2}a{t}^2+vt+s$  
  •    h    = final height  
  •    t    = time  
  •    v    = initial velocity  
  •    s    = starting height  
  •    a    = acceleration due to gravity
• Depending on the    direction    of the parabola and the    measurements    used, $a=$ acceleration due to gravity will be one of these options:
  • 32 feet per second squared ($32 \raisebox{1ex}{$\mathrm{ft}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{s}}^2$}\right.$)
  • 9.8 meters per second squared ($9.8 \raisebox{1ex}{$\mathrm{ft}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{s}}^2$}\right.$)
• Remember, creating a    sketch    of a word problem can help you determine what is happening in the problem.

Example 5

A football player caught a ball kicked downfield. The catch was made 3 feet above the ground. If the ball was kicked with an initial velocity of 75 feet per second, how much time passed before the ball was caught? (Hint: The ball was kicked from the ground.)

Note

The answer of 0.040 seconds does not make sense for this problem because the ball cannot be caught downfield in less than one second. 


 

Q: What is the start and end height of the ball for this scenario? Explain.

A: The start height is 0 feet because the ball is kicked from the ground. The end height is 3 feet because the ball was caught by the football player at that height.

Example 6

In an egg drop experiment, the goal is to cushion an egg so that when it is dropped from a height of 50 feet, it does not crack. The science teacher starts by having the students calculate the drop time of the control group (an uncushioned egg). Determine the time a control group egg takes to hit the ground.