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Homework: Introduction to Quadratic Equations

Module by: Kenny M. Felder. E-mail the author

If a ball is thrown up into the air, the equation for its position is: h ( t ) = h o + v o t 16 t 2 h ( t ) = h o + v o t 16 t 2 size 12{h \( t \) =h rSub { size 8{o} } +v rSub { size 8{o} } t - "16"t rSup { size 8{2} } } {} where…

  • hh size 12{h} {} is the height—given as a function of time, of course—measured in feet.
  • tt size 12{t} {} is the time, measured in seconds.
  • hoho size 12{h rSub { size 8{o} } } {} is the initial height that it had when it was thrown—or, to put it another way, hoho size 12{h rSub { size 8{o} } } {} is height when t=0t=0 size 12{t=0} {}.
  • vovo size 12{v rSub { size 8{o} } } {} is the initial velocity that it had when it was thrown, measured in feet per second—or, to again put it another way, vovo size 12{v rSub { size 8{o} } } {} is the velocity when t=0t=0 size 12{t=0} {}.

This is sometimes called the equation of motion for an object, since it tells you where the object is (its height) at any given time.

Use that equation to solve the questions below.

Exercise 1

I throw a ball up from my hand. It leaves my hand 3 feet above the ground, with a velocity of 35 feet per second. (So these are the initial height and velocity, hoho size 12{h rSub { size 8{o} } } {} and vovo size 12{v rSub { size 8{o} } } {}.)

  • a.: Write the equation of motion for this ball. You get this by taking the general equation I gave you above, and plugging in the specific hoho size 12{h rSub { size 8{o} } } {} and vovo size 12{v rSub { size 8{o} } } {} for this particular ball.
  • b.: How high is the ball after two seconds? (In other words, what hh size 12{h} {} value do you get when you plug in t=2t=2 size 12{t=2} {}?)
  • c.: What hh size 12{h} {} value do you get when you plug t=0t=0 size 12{t=0} {} into the equation? Explain in words what this result means.

Exercise 2

I throw a different ball, much more gently. This one also leaves my hand 3 feet above the ground, but with a velocity of only 2 feet per second.

  • a.: Write the equation of motion for this ball. You get this by taking the general equation I gave you above, and plugging in the specific hoho size 12{h rSub { size 8{o} } } {} and vovo size 12{v rSub { size 8{o} } } {} for this particular ball.
  • b.: How high is the ball after two seconds? (In other words, what hh size 12{h} {} value do you get when you plug in t=2t=2 size 12{t=2} {}?)
  • c.: What hh size 12{h} {} value do you get when you plug t=0t=0 size 12{t=0} {} into the equation? Explain in words what this result means.

Exercise 3

A spring leaps up from the ground, and hits the ground again after 3 seconds. What was the velocity of the spring as it left the ground?

Exercise 4

I drop a ball from a 100 ft building. How long does it take to reach the ground?

Exercise 5

Finally, one straight equation to solve for xx size 12{x} {}:

( x 2 ) ( x 1 ) = 12 ( x 2 ) ( x 1 ) = 12 size 12{ \( x - 2 \) \( x - 1 \) ="12"} {}

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