Quadratic Application: Projectile Motion

Quadratic Applications: Projectile Motion 

Generally speaking, projectile motion problems involve objects that are thrown, shot, or dropped. Usually the object will be launched directly upward or dropped directly down.

Consider the following example:

An object is launched directly upward at 19.6 m/s from a 58.8-meter tall platform.  The equation for the object's height (s) at time (t) seconds after launch is s(t) = -4.9t2 + 19.6t + 58.8, where s is in meters.  When does the object strike the ground?


This question asks for the time when the object strikes the ground.  Another way of thinking of this is, when is the object's height zero (on the ground)?  So, set s equal to zero and solve the equation:

0 = -4.9t2 + 19.6t + 58.8
0 = t2 - 4t - 12
0 = (t - 6)(t + 2)
t = 6 or t = -2

Recall that t is a value of time, answering the question "when."  It doesn't make sense in this context for t to be a negative value, so t = -2 is an extraneous solution.

The object strikes the ground 6 seconds after launch.

Formula of Projectile Motion Problems

The key features to know in projectile motion problems are the initial height, the initial speed, a
value of the force of gravity, and time.  They are related in the formula given below:

s(t) = -gt2 + v0t + h0


  • s  is the height at any particular time (t)  [Note:  s(t) is also sometimes shown in the formula as h]
  • g is gravity value – in feet this value is 16 and in meters this value is 4.9  [Note:  In physics, the gravitational constant is actually 32 for feet and 9.8 for meters, but the formula uses one-half this value.]
  • v0 is the initial velocity 
  • h0 is the initial height