**Introduction to Finding 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* = – 16 *t* ^{2} + *v* _{0} *t* + *h* _{0} , where *h* is the object’s height (in feet), *t* is time (in seconds), *h* _{0} is the object’s initial height (that is, its height at *t* = 0 seconds) and *v* _{0} 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, *v* _{0} is positive. If the object is thrown downward, *v* _{0} is negative. If the object is dropped, *v* _{0} 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?

**Determining When an Object Will Reach the Ground **

**Examples**

**Example 1:**

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, *v* _{0} , is zero: *v* _{0} = 0. The object is dropped from a height of 1600 feet, so *h* _{0} = 1600. The formula *h* = –16 *t* ^{2} + *v* _{0} *t* + *h* _{0} becomes *h* = –16 *t* ^{2} + 1600. The object hits the ground when *h* = 0, so *h* = –16 *t* ^{2} + 1600 becomes 0 = –16 *t* ^{2} + 1600.

The object will hit the ground 10 seconds after it is dropped.

**Example 2:**

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, *v* _{0} , is zero: *v* _{0} = 0. The object is dropped from a height of 48 feet, so *h* _{0} = 48. The formula *h* = –16 *t* ^{2} + *v* _{0} *t* + *h* _{0} becomes *h* = –16 *t* ^{2} + 48. The object hits the ground when *h* = 0.

The ball will reach the ground in about 1.73 seconds.

Find practice problems and solutions at The Height of a Falling Object Practice Problems - Set 1.

**Determining When an Object Will Reach a Specific Height**

**Example**

An object is dropped from the roof of a 60-foot building. How long must it fall to reach a height of 28 feet?

In the formula *h* = –16 *t* ^{2} + *v* _{0} *t* + *h* _{0} , *h* _{0} is 60 and *v* _{0} is zero (because the object is dropped). The object reaches a height of 28 feet when *h* = 28.

The object will reach a height of 28 feet after about 1.41 seconds.

Find practice problems and solutions at The Height of a Falling Object Practice Problems - Set 2.

**Determing How Long or at What Time an Object Will Reach a Specific Height**

**Examples**

**Example 1:**

An object is tossed up in the air at the rate of 40 feet per second. How long will it take for the object to hit the ground?

In the formula *h* = –16 *t* ^{2} + *v* _{0} *t* + *h* _{0} , *v* _{0} = 40 and *h* _{0} = 0.

*h* = –16 *t* ^{2} + 40 *t*

We want to find *t* when *h* = 0.

The object will hit the ground after 2.5 seconds.

**Example 2:**

A projectile is fired upward from the ground at an initial velocity of 60 feet per second. When will the projectile be 44 feet above the ground?

In the formula *h* = –16 *t* ^{2} + *v* _{0} *t* + *h* _{0} , *v* _{0} = 60 and *h* _{0} = 0.

*h* = –16 *t* ^{2} + 60 *t*

We want to find *t* when *h* = 44.

The projectile will be 44 feet off the ground at 1 second (on the way up) and again at 2.75 seconds (on the way down).

Find practice problems and solutions at The Height of a Falling Object Practice Problems - Set 3.

More practice problems for this concept can be found at: Algebra Quadratic Applications Practice Test.

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