Quadratic Application: 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
- 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
1) An object is launched from ground level directly upward at 39.2 m/s. For how long is the object at or above a height of 34.3 meters?
Step 1: Recall the formula: s(t) = -g2 + v0t + h0
Step 2: Use the gravity value in meters, insert the initial velocity and heights given: s(t) = -4.9t 2 + 39.2t + 0
Step 3: We need the times when the height s(t) = 34.3 m
Step 4: Solve -4.9t 2 + 39.2t = 34.3 and subtract the two answers.
2) Two objects are dropped from a bridge 160 feet above ground level. One object is released with an initial velocity of zero, falling only by the force of gravity. The other is thrown straight down with a velocity of 48 ft/s. How many seconds earlier does the thrown object reach the ground as compared with the dropped object?
thrown object: s(t) = -16t 2 − 48t + 160
3) An object is thrust directly down from a height of 190 feet at a velocity downward of 64 ft/s. How many seconds will it take before the object hits the ground?
1) 6 seconds
2) 1.16 seconds
3) 5.98 seconds