Motion Problems
Here a great diagram that represents the relationships between the variables in motion problems.
The D stands for distance. The R stands for rate or velocity. The T stands for time. It is easy to see the relations from the diagram.
Rate x Time = Distance
Distance/Time = Rate
Distance/Rate = Time
Relative Rates
On occasion objects move within a medium which is moving with respect to an observer. For example, if we are going down the highway at 60 mph and someone overtakes us by going 10mph faster than us, the relative rate is 10mph. However, if we are traveling down the highway at 60mph and someone on the other side of the highway is going in the opposite direction at 70mph, the relative rate would be as if we were coming together at 130mph.
Example:
One hour after Yolanda started walig from X to Y, a distance of 45 miles, Bob started walking along the same road from Y to X. If Yolanda’s walking rate was 3 miles per hour and Bob’s was 4 miles per hour, how many miles has bBob walked when they met?
(A) 24
(B) 23
(C) 22
(D) 21
(E) 19.5
Solution:
We can make a quick diagram of the problem.
One way to solve this problem is with a table. We want to know the distance Bob walked. The distance Bob walked is our variable.
Let x = the distance Bob walked.
Together Yolanda and Bob walked 45 miles, so Yolanda walked 45 -x miles when they met. We are given the rate at which both of them walk. We are told Yolanda walked 1 more hour than Bob. If Bob walked t,Yolanda walked t + 1. The results are summarrized in the table below.
We get two equations:
i)x = 4t
ii)45 - x = 3t + 3
We can substitute the value of x in i) into ii)
45 - 4t = 3t + 3
7t = 42
t = 6
Since we are looking for x, we substitute t back into i)
x = 4t = 4 x 6 = 24
The correct answer is A.
There is a much faster way to solve this problem.
Yolanda started walking for an hour at 3mph, so after an hour she walk 3 miles. The distance they have to now cover is 42 miles.
Now Bob starts walking and Yolanda and Bob are walking toward each other at a relative rate of 3+ 4= 7 mph. They have to cover 42 miles walking at a combined rate of 7mph.
D = R x T
42 = 7t
t = 6
It will take them 6 hours to meet.
Bob’s distance= R x T = 4 x 6 = 24
Example:
A car travels from M to R at an avgerage speed of 30mph and returns along the same route at an average speed of 40 miles per hour. Which is the closest average speed in mph for the roundtrip?
(A) 32.0
(B) 33.0
(C) 34.3
(D) 35.5
(E) 36.5
Solution:
This is a typical motion problem. We can not just add the rates and divide by 2. This would be wrong. Instead, we set up a table.
Remember our basic relations that we can see if we draw the triangle noted above.
Rate x Time = Distance
Distance/Time = Rate
Distance/Rate = Time
Our journey consists of two parts from M to R and then from R to M.
We are looking for the average rate. This is the rate when we travel the whole distance.
R = D/T
The correct answer is C.
(46)
(28)
