Distance Problems: Uniform Motion
Distance or Uniform Motion problems are worked in two parts. First the distance formula is used, usually in chart form, to find the individual distances. Then a general equation is set up, based on the type of problem given, and then the equation is solved.
Section A: One object traveling in one direction
I.One object moving: Distance Formula: d = r t (distance = rate x time)
a. A bird flies at a speed of 8 mph for 2 hours. How far did the bird travel?
d = (8)(2)
d= 16 miles
b. If a bird flies at a speed of 10 mph, how long would it take for the bird to travel a distance of 12 miles?
(12) = (10)t
12/10 = t
1.2 hours = t
Time conversion: .2 hours to minutes
.2 x 60 = 12 minutes
So the bird took 1 hour 12 minutes to make the trip.
c. If a bird flies 17 miles in 3 hours, what was its speed?
(17) = r (3)
17/3 = r
5.66…
Round off to one decimal—rate (speed) = 5.7 miles per hour
Section B: Two objects traveling in a straight line (towards or away from each other)
- Organize the given information (usually in The Distance Chart)
- Choose the Correct formula:
- Formula: Distance of 1st train + Distance of 2nd train = Total Distance
- Formula: TotalDistance - Distance of (either) train = Distance of (the other) train
- Formula:
Note: Formula c is the Pythagorean theorem which is used if objects are moving at right angles (i.e. one is going east and the other is going north)
EXAMPLES:
Two Trains : Two trains(cars, planes, etc.) travel in a straight line
- Identification Key: Total distance is always given.
Version one: Two vehicles start in the same place and head away from each other.
Version two: Two vehicles start in different places and head towards each other.
-Use Formula a if both individual distances are given
-Use Formula b if one individual distance is given and you need to find the other one.
-Use Formula c for objects traveling at right angles:
EXAMPLES:
Catch-Up: Two people (cars, planes, etc.).
Identification Key: Total distance is NOT given.
3 versions: Unknown TIME, DISTANCE, or RATE
- Time problem: One person jogs down a trail at 2 mph for 1 mile. Then a second one starts along the same trail traveling at 3 mph. The second one passes the first at the 6 mile marker. How long did it take for the second one to catch-up with the first (or to pass the first).
- Calculate Individual Times (d=rt):
Note: The first runner’sdistanceof one mile is a distractor. They both had to travel that same mile for one to catch up with the other one. At the 6 mile marker, they have both traveled the same distance, a total of 6 miles.
Runner 1: 6 = (2)t, (divide by 2)
(time of runner 1) = 3 hours
(i.e. the first runner had been running for 3 hours when the 2nd passed him)
Runner 2:6 = (3)t, (divide by 3)
(time of runner 2) = 2 hours
(i.e. the 2nd runner had been running for 2 hours when he passed the 1st runner)
2. Answer: The question asked for the time of the 2nd runner, so it is 2 hours
- Distance problem: One person jogs down a trail at 4 mph, then a second one starts along the same trail and jogs at a speed of 6 mph. How far did they travel if it took 2 hours for the second one to pass (or catch up with) the first one?
- Calculate Individual Distances (d=rt):
Runner 1: d = (3)(2+x), d1= 6+3x miles
Runner 2: d = (4)(x), d2= 4x miles
- They have traveled the same distance when one catches up with the other, so using the distance formula with their individual distances: d1=d2
d1 = d2
6 + 3x = 4x (subtract 3x from both sides)
6 = x
Answer: One passed the other one at the 6 mile marker.
- Rate problem: One person jogs down a trail for 1 hour. A short time later a second one starts along the same trail. After 1/2 an hour, the second one passes the first at the 3 mile marker. How fast was each one traveling?
- Calculate Individual Rates (d=rt):
Runner 1: (3) = r (1+.5) 3 = 1.5r (divide by 1.5) r1 = 2 mph
Runner 2: (3) = r (.5) 3 = .5r (divide by .5) r2 = 6 mph
- Answer: The question asks for rates of both of them, so runner 1 has a rate of 2 mph and runner 2 has a rate of 6 mph.