The Height of a Falling Object
The height of an object dropped, thrown or fired can be computed using quadratic equations. The general formula is h = –16t2 + v0t + h0, where h is the object’s height (in feet), t is time (in seconds), h0 is the object’s initial height (that is, its height at t = 0 seconds) and v0 is the object’s initial velocity (that is, its speed at t = 0 seconds) in feet per second. If the object is tossed, thrown, or fired upward, v0 is positive. If the object is thrown downward, v0 is negative. If the object is dropped, v0 is zero. The object reaches the ground when h = 0. (The effect of air resistance is ignored.)
Typical
questions are:
When
will the object be ___ feet high?
When
will the object reach the ground?
What
is the object’s height after ____ seconds?
Examples
An object is dropped from a height of 1600
feet. How long will it take for the object to hit the ground?
Because
the object is dropped, the initial velocity, v0, is zero: v0
= 0.
The object is dropped from a height of 1600
feet, so h0 = 1600. The formula h = –16t2 + v0t + h0 becomes h = –16t2 + 1600. The object hits the
ground when h = 0, so h = –16t2
+ 1600 becomes 0 = –16t2
+ 1600.
The object will hit the ground 10 seconds
after it is dropped.
A ball is dropped from the top of a
four-story building. The building is 48 feet tall. How long will it take for
the ball to reach the ground?
Because
the object is dropped, the initial velocity, v0, is zero: v0
= 0. The object is dropped from a height of 48 feet, so h0 = 48. The formula h =
-16t2 + v0t + h0 becomes h = –16t2 + 48. The object hits the ground
when h = 0.
The ball will reach the ground in about 1.73
seconds.
Practice
1. An
object is dropped from a 56-foot bridge over a bay. How long will it take for
the object to reach the water?
2. An
object is dropped from the top of a 240-foot tall observation tower. How long
will it take for the object to reach the ground?
3. A ball
is dropped from a sixth-floor window at a height of 70 feet. When will the ball
hit the ground?
4. An
object falls from the top of a 100-foot communications tower.After how much
time will the object hit the ground?
Solutions
For all of these problems, both a negative t and
a positive t will be solutions for the quadratic equations. Only the positive t
will be a solution to the problem.
1. For the
formula h = –16t2 + v0t + h0, h0 = 56 and v0 = 0 (because the
object is being dropped). The object reaches the ground when h = 0.
The object will reach the water in about 1.87
seconds.
2. For the
formula h = –16t2 + v0t + h0, h0 = 240 and v0
= 0 (because the object is being dropped). The object reaches the ground when h
= 0.
The object will reach the ground in about
3.87 seconds.
3. For the
formula h = –16t2 + v0t + h0, h0 = 70 and v0
= 0 (because the object is being dropped). The object reaches the ground when h
= 0.
The ball will hit the ground in about 2.09
seconds.
4. For the
formula h = –16t2 + v0t + h0, h0 = 100 and v0
= 0 (because the object is being dropped). The object reaches the ground when h
= 0.
The object will hit the ground after 2.5
seconds.
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