Many algebraic word problems involve fundamental physical relationships. Among the most common are problems involving motion. For example, the definition of speed is distance traveled divided by the time it takes. or multiplying both sides by t, d = V_ave x t. For example, if a car travels at 50 miles per hour for 2 hours, the distance traveled equals (50 mi/hr)(2 hr) or 100 miles. This relationship applies for constant velocity motion only. In practice, it is applied more generally by using an average speed or average rate of travel for the time involved. The distance traveled is often represented by s; the average speed or average rate of travel, also called the average velocity, by v_av; and the time of travel by t.

This same basic physical relationship can be written in two other forms, obtained by dividing both sides of the equation by v_av or by t.

Example 1:

How far can a car traveling at a rate of 52 miles per hour travel in 2^1/2 hours? Solution:

Using Equation 2-13:

Example 2:

How long does it take a plane traveling at 650 miles per hour to go 1430 miles? Solution:

Using Equation 2-14:

Example 3:

What is the average speed of a train that completes a 450-mile trip in 5 hours?

Solution:

Using Equation 2-15:

Algebraic word problems involving motion are solved using the general relationship among distance, time, and average velocity following the same five basic steps for solving any algebraic word problem.

Example 1:

A plane flying at 525 miles per hour completes a trip in 2 hours less than another plane flying at 350 miles per hour. What is the distance traveled?

Solution:

Step 1.Let x = Distance Traveled (in miles)

Step 2.Then, using Equation 2-14,

Step 3.Time Taken by Faster Plane = Time Taken by Slower Plane - 2 hours

Solving for the other unknowns:

= Time Taken by Faster Plane (in hours)

= Time Taken by Slower Plane (in hours)

Answers:

Distance Traveled = 2100 miles

Time Taken by Faster Plane = 4 hours

Time Taken by Slower Plane = 6 hours

Step 5.The faster plane takes 2 hours less to complete the trip than the slower plane.

6 hours - 2 hours = 4 hours

Thus, the answer checks.

Example 2:

It takes a man 4 hours to reach a destination 1325 miles from his home. He drives to the airport at an average speed of 50 miles per hour, and the average speed of his plane trip is 500 miles per hour. How far does he travel by each mode of transportation?

Solution:

Step 1.Let x = Distance Traveled by Car (in miles)

Step 2.Then,

1325 - x = Distance Traveled by Plane (in miles) and, using Equation 2-14,

Step 3.Total Time = (Time Traveled by Car) + (Time Traveled by Plane)

Step 4.Solving for x:

Solving for the other unknowns:

Answers:

Distance Traveled by Car = 75 miles

Distance Traveled by Plane = 1250 miles

Step 5.The total distance traveled is 1325 miles.

75 miles + 1250 miles = 1325 miles

The average speed by car is 50 miles per hour.

The average speed by plane is 500 miles per hour.

The total time traveling is 4 hours.

Thus, the answers check.