Motion Under Gravity

Date: 07/18/2003 at 22:30:38
From: Jen
Subject: Algebra

Motion Under Gravity:  Use the equation 

   s = -16t squared + vt 

to determine when the height of an object is 960 feet if the initial 
velocity is 272 feet per second. Solve either algebraically or 
graphically. Include all algebra to justify response if solving 
algebraically. Include sketch of graph and explain how you interpreted 
the graph to get your answer(s).

Date: 07/21/2003 at 12:56:42
From: Doctor Ian
Subject: Re: Algebra

Hi Jen,

The equation is

  s = -16t^2 + vt

Let's think about what this means. It means that at time zero, the
height is also zero. So the object is starting out from the ground,
or wherever we're measuring height from.  

Now, suppose there were no gravity. Then we could figure out how high
the object is by multiplying the velocity by the time. This is the
same thing we do when we note that at 60 miles per hour, a car travels
60 miles in one hour, 120 miles in two hours, and so on. 

But there _is_ gravity, and that's going to accelerate the object back
toward earth. That's why it has a negative sign, because it's working 
in a direction opposite to the initial velocity.  Note that as t 
increases, the size of the acceleration term will eventually become
larger than the size of the velocity term (because t^2 will eventually
be much larger than t), so eventually the object will end up returning
to the ground.  

So that's what the equation is saying, albeit in a very compact way.

You're told that the initial velocity is 272 feet/sec.  So you can
substitute that for v:

  s = -16t^2 + 272t

And you're told that at some point, the height will be 960 feet:

                 960 = -16t^2 + 272t

which means that 

  16t^2 - 272t + 960 = 0

And dividing both sides of the equation by 16 gives us

      t^2 - 17t + 60 = 0

So this is a quadratic equation with variable t.  You can try to
factor it, or you can use the quadratic formula to get the answer
directly. 

Is this enough to get started? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/