Motion Problems

Cars travel in opposite directions at 56 MPH and 60 MPH. When will the cars be 405 miles apart?

Rate / Time / Distance
Slow Car / 56 / H / 56H
Fast Car / 60 / H / 60H

56H+60H=464; 116H=464; H=4

1.  Example 2

2 Trains from 405 miles apart travel toward each other. One train is going twice the rate of the other. In 3 hours they pass each other. How fast are they going? Draw a diagram

Rate / Time / Distance / Answer
Freight Train / R / 3 / 3R / 3R+3(2R)=9R=405; R=45
Passenger Train / 2R / 3 / 3(2R)

2.  Example 3

2 Trains start toward each other from 360 miles apart. The rate of train 1 exceed the rate of train 2 by 10 miles. After 2 hours the trains are 120 miles apart. Find the rate of each.

Rate / Time / Distance / Answer
Slow Train / R / 2 / 2R / 2R+2(r+10)+120=360
2R+2R+20+120=360; 4R+140=360; 4R=220; R=55
Fast Train / R+10 / 2 / 2(r+10)

Individual Activities:

1)  Two planes started at the same time from the same airport in opposite directions. 1 flew 60 miles faster than the other. In five hours they were 5300 miles apart. Find the rate.

Rate / Time / Distance / Answer
Plane 1 / R+60 / 5 / 5(r+60) / 5R+5(r+60)=5300;5R+5R+300=5300; 10R=5000 R=500
Plane 2 / R / 5 / 5R

2) Two trains started from the same station and traveled in opposite directions. After 10 hours they were 1400 miles apart. The rate of the first exceeded the rate of the slow train by 5 miles per hour. Find the rate of each train

Rate / Time / Distance / Answer
Slow Train / R / 10 / 10R / 10R+10R+50=1400; 20R=1350 R=67.50
Fast Train / R+5 / 10 / 10(r+5)

3)At 6AM 2 planes started from the same airport and flew west. Plane 1 averaged 520 MPH and the other 600 MPH. When were the planes 280 miles apart? Represent graphically

Rate / Time / Distance / Answer
Plane 1 / 520 / T / 520 T / 520T+600T= 280; 80T=280 ; T=3.5hours Time =9:30
Plane 2 / 600 / T / 600T

Worksheet (Vehicles traveling in Opposite Directions)

Model Example Two planes started at the same time from the same airport in opposite directions. 1 flew 60 miles faster than the other. In five hours they were 5300 miles apart. Find the rate.

Rate / Time / Distance / Answer
Plane 1 / R+60 / 5 / 5(r+60) / 5R+5(r+60)=5300;5R+5R+300=5300; 10R=5000 R=500
Plane 2 / R / 5 / 5R

Additional Problems:

1.  Maria and Monique start out at 10AM at points which are 35 miles apart and travel toward each other. Maria walks at the rate of 4 m.p.h., while Monique roller blades at the rate of 10 m.p.h.. At what time will they meet?

Rate / Time / Distance / Answer
Maria / 4 / X / 4x
Monique / 10 / x / 10x

2.  Two cars leave from 116th Street traveling in opposite directions with one car averaging 4 m.p.h. more than the other. After 2 hours, the cars are 212 miles apart. Find the rate of each car.

Rate / Time / Distance / Answer
Car 1 / X / 2
Car 2 / X+2 / 2

3.  Justin left home at 8:00 a.m. riding his bike at 6 m.p.h.. His brother Joshua left 2 hours

later, riding his bike at 8 m.p.h.. At what time will Joshua catch up to Justin if Joshua is

delayed 15 minutes by a freight train?

Rate / Time / Distance / Answer

4.  Two planes travel in opposite directions at 300 m.p.h. and 500 m.p.h. In how many hours

will they be 2,000 miles apart?

Rate / Time / Distance / Answer

5.  Rashid and Hasan left home at the same time. Rashid walked toward the Park at the rate of 3 m.p.h. In how many hours will Rashid and Hasan be 15 miles apart?

Rate / Time / Distance / Answer

6.  Two ships are at ports 310 miles apart start sailing towards each other at 11:00 a.m.. If their average rates are 22 mph and 34 mph, at what time will they be 30 miles apart?

Rate / Time / Distance / Answer

Problems:

1.  A plane left Kennedy Airport at 5:00 a.m. traveling east at the rate of 550 m.p.h. At 7:00 a.m. another plane left the same airport traveling at the rate of 825 m.p.h. At what time does the second plane overtake the first?

2.  Contrast this problem with the problem above: At 5:00 a.m., two trains leave the same city and travel in opposite directions. Their rates differ by 5 m.p.h. At 11:00 a.m. the trains are 450 miles apart. Find the rate of each train. How far has each train traveled?

3.  A steamship left San Francisco for Hawaii traveling at the rate of 50 m.p.h. Six hours later a plane is sent out to overtake the steamship. If the plane’s speed is 300 m.p.h., how long will it take for the plane to catch up with the steamship? How far will they be from San Francisco?

4.  Mary left home walking toward a park at the rate of 3 m.p.h. John left an hour later. If he walks at the rate of 4 m.p.h. In how many hours will he catch up with Mary?

5.  At 10:00 a.m. two cars started at the same point and traveled in the same direction. One car averaged 53 m.p.h. while the other averaged 44 m.p.h. In how many hours will the two cars be 36 miles apart?

6.  Jack left one end of a lake to row to the opposite end at the same time that Jill left the opposite end to meet Jack. If Jack rowed 3 m.p.h. while Jill rowed 2 m.p.h., how long will it take for them to meet if the distance between them was initially 4 miles?

7.  Ben, who is bicycling at the rate of x mph, had a head start of 2 miles on Rachel, who is bicycling at the rate of 4 mph (y x). In how many hours will Rachel overtake Ben?