Mrs. Napholtz's Math Site

Motion Problems

An airplane takes 2 h 30 min to travel 500 km against the wind. With the same wind, the return trip takes 1 h 40 min. Find the rate of the wind.

2.5(p - w) = 500 p - w = 200
5/3(p + w) = 500 p + w = 300
2p = 500
p = 250
w = 50
ANS: The speed of the wind is 50 km/hr.

In a canoe race, a team paddles downstream 560 m in 70 s. The same team makes the trip upstream in 80 s. Find the team's rate in still water and the rate of the current.

70(b + c) = 560 b + c = 8
80(b - c) = 560 b - c = 7
2b = 15
b = 7.5
c = 0.5
Ans: The team's rate in still water is 7.5 m per second.
The rate of the current is 0.5 m per sec.

A plane flying with the wind traveled 1260 km in 2 h. On the return trip against the same wind, the trip took 3 h. Find the rate of the plane in still air, and the rate of the wind.

(p + w) * 2 = 1260
(p - w) * 3 = 1260    Simplify these equations.
p + w = 630
p - w = 420    Add these two equations.
2p = 1050
p = 525
w = 105

A bicyclist rode up a mountain road at 12 km/h and then back down at 30 km/h. If the round trip took 3.5 hours, how long did the ride up the mountain take?

The distance up the mountain = the distance down the mountain.
12n = 105 - 30n
42n = 105
n = 2.5 hours.

A plane leaves New York for San Francisco and averages 800 km/h. At the same time a plane leaves San Francisco for New York and averages 880 km/h. If the two cities are 5040 km apart, in how many hours will the planes pass each other?

The distances must equal 5040
800n + 880n = 5040
1680n = 5040
n=3
Answer: The planes will pass each other after 3 hours.

Carlos begins jogging west at 12 km/hr at 10:00 A.M. Denise leaves the same place at 11:00 A.M. and bicycles at 36 km/hr toward Carlos. At what time of day will she overtake him?

Note: Denise travels one hour less than Carlos!
When Denise catches up, Carlos and Denise will have traveled equal distances.
12n = 36n - 36
-24n = -36
n = 1.5
Answer: Denise overtakes Carlos at 11:30 A.M.

A bus takes 12 hours to travel the same distance that an airplane travels in 1.5 hours. The average speed of the airplane is 120 km per hour faster than five times the average speed of the bus. What is the average speed of the bus?

The problem clearly states that the bus and the airplane travel equal distances.
12n = 7.5n + 180
4.5n = 180
n = 40
Answer: The bus travels at 40 km/h.

Sylvia walked to Cathy's house at 2 km/h, and then rode a bicycle back to her own house at 10 km/h. The entire trip took 3 hours. How long a time did Sylvia walk?

Since the lengths of the trips are equal, we get
2n = 30 - 10n
12n = 30
n = 2.5
Answer: Sylvia walked for 2 and a half hours.

At 7:00 A.M. Joe starts jogging at 6 mph. At 7:10 A.M. Ken starts off after him. How fast must Ken run in order to overtake him at 7:30 A.M.?

Before clicking on animated demonstration, think:
Joe will have jogged for 30 minutes = 1/2 hour.
Ken will have jogged for 20 minutes = 1/3 hour.

Click here for animated solution.

It takes a plane 40 min. longer to fly from Boston to Los Angeles at 525 mph than it does to return at 600 mph. How far apart are the cities?

Assume the flight paths are of equal length:
(Distance from Boston to LA) = (Return trip distance)
525n + 350 = 600n
350 = 75n
14/3 = n
Using this value of n, we can compute the length of either trip: Return trip = 600(14/3) = 2800 miles. ANSWER: The cities are 2800 miles apart.

A bus traveled 387 km in 5 hours. One hour of the trip was in city traffic. The bus's city speed was just half of its speed on open highway. The rest of the trip was on open highway. Find the bus's city speed.

Since the total trip ws 387 km long,
(Distance in city traffic) + (Distance on highway) = 387
(n/2) + (4n) = 387
4.5n = 387
n = 387/4.5 = 86 km/h.
ANSWER: The bus's city speed was n/2 = 43 km/h

Two jets leave Ontario at the same time, one flying east at a speed 20 km/h greater than the other, which is flying west. After 4 hours, the planes are 6000 km apart. Find their speeds.

After 4 hours, the jets are 6000 km apart, which means
(Distance East) + (Distance West) = 6000
(4n + 80) + (4n) = 6000
8n + 80 = 6000
8n = 5920
n = 740 km / h
ANSWER: West: 760 km/h; East: 740 km/hr

Two camper vans leave Arrowhead Lake at the same time, one traveling north at a speed of 10 km/h faster than the other, which is traveling south. After 3 hours, the camper vans are 420 km apart. Find their speeds.

After 3 hours, the vans are 420 km apart, which means
(Distance North) + (Distance South) = 420
(3n + 30) + (3n) = 420
6n + 30 = 420
6n = 390
n = 65 km/h
ANSWER: South: 65 km/h; North: 75 km/hr

Two cars traveled in opposite directions from the same starting point. The rate of one car was 10 km less than the rate of the other. After 4 hours the cars were 600 km apart. Find the rate of each car.

After 4 hours, the vans are 600 km apart, which means
(Distance of One Car) + (Distance of Other Car) = 600
(4n - 40) + (4n) = 600
8n - 40 = 600
8n = 640
n = 80 km/h
ANSWER: "Other car: 80 km/h; "One car": 70 km/hr

Two trains leave Dallas at 9 A.M. The eastbound train travels at 45 mph. The westbound train travels at 60 mph. At what time will the two trains be 420 miles apart?

Since the trains are moving in opposite directions, the distance between them at any time is 45n + 60 n.
45n + 60n = 420
105n = 420
n = 4
Four hours after 9 A.M. is 1:00 P.M.
Answer: The trains will be 420 miles apart at 1:00 P.M.

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