Relative Motion and Catch-Up Scenarios Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB)

Question 1Question →

Two cyclists, Sarah and Mike, start at opposite ends of a $60$ -mile trail and ride directly toward each other. Sarah rides at a constant speed of $12$ mph, and Mike rides at a constant speed of $18$ mph. How many hours will it take for them to meet?

Show answer & explanation

Answer: 2

Answer

2

Because the cyclists are moving directly toward each other, their gap closes at a combined rate of

30

mph (

12

mph +

18

mph). Dividing the total initial distance of

60

miles by this combined rate yields exactly

2

hours.

Step-by-Step Solution

1

Calculate the combined relative speed.

30 mph

When objects move toward each other from opposite directions, their relative speed is the sum of their individual speeds (

12 + 18 = 30

).

2

Calculate the time required to meet.

2 hours

Using the formula Time = Distance / Rate, divide the total distance (

60

miles) by the relative speed (

30

mph).

Key Concept

Relative Motion and Catch-Up Scenarios

Question 2Question →

Two cars leave a parking lot at the same time and travel in opposite directions on a straight highway. One car travels at a constant speed of $40$ mph, and the other car travels at a constant speed of $50$ mph. How many hours will it take for the two cars to be exactly $270$ miles apart?

3

6

27

30

Show answer & explanation

Answer: 3

Answer

It will take 3 hours for the cars to be 270 miles apart.

The correct answer is 3. Since the cars are traveling in opposite directions, their speeds are added to find the total rate at which they are separating:

40 \text{ mph} + 50 \text{ mph} = 90 \text{ mph}

. To find the time it takes to separate by

270

miles, divide the total distance by the combined relative speed:

270 \div 90 = 3

hours.

Step-by-Step Solution

1

Determine the relative speed of the two cars.

40 + 50 = 90

mph

When two objects move in opposite directions away from each other, their relative speed is the sum of their individual speeds.

2

Calculate the time required to reach the target distance.

270 \div 90 = 3

hours

Time equals total distance divided by the relative speed (

T = D \div R

).

Key Concept

Relative Motion (Opposite Directions)

Question 3Question →

Two inspection drones take off from the same cellular tower at the same time and fly in opposite directions along a straight power line. Drone A flies at $15$ mph and Drone B flies at $25$ mph. How many miles apart are the two drones after $2$ hours?

Show answer & explanation

Answer: 80; 80 miles; 80 mi; eighty

Answer

The drones are 80 miles apart.

Since the drones travel in opposite directions, their relative speed is the sum of their individual speeds (

15 + 25 = 40

mph). Over the course of

2

hours, the distance between them becomes

40 \times 2 = 80

miles.

Step-by-Step Solution

1

Calculate the combined relative speed of both drones.

15 + 25 = 40

mph

Because the drones are flying in opposite directions, the distance between them increases at a rate equal to the sum of their individual speeds.

2

Multiply the relative speed by the time traveled.

40 \times 2 = 80

miles

Total distance is found by multiplying the relative speed by the total time (

D = r \times t

).

Key Concept

Calculating separation distance for objects moving in opposite directions using relative speed.

Estimated Time:45s

Question 4Question →

Two motorboats leave a marina at the same time, traveling in opposite directions on a straight path. One boat travels at a constant speed of $18$ mph, and the other travels at a constant speed of $22$ mph. How many minutes will it take for the two boats to be exactly $120$ miles apart?

3

180

360

1800

Show answer & explanation

Answer: 180

Answer

180

Because the boats are moving in opposite directions, their relative speed of separation is the sum of their individual speeds: 18 mph + 22 mph = 40 mph. To find how long it takes to separate by 120 miles, divide the distance by the relative rate: 120 / 40 = 3 hours. Finally, convert 3 hours into minutes by multiplying by 60, resulting in 180 minutes.

Step-by-Step Solution

1

Calculate the combined relative speed of the two boats.

18 + 22 = 40

mph

Because the boats are traveling in opposite directions, their speeds add together to create the rate at which they are separating.

2

Calculate the time in hours it takes to reach the target distance.

120 \div 40 = 3

hours

Using the formula Time = Distance / Rate, dividing the total distance by the relative speed gives the time in hours.

3

Convert the time from hours to minutes.

3 \times 60 = 180

minutes

The question explicitly asks for the final answer in minutes, not hours.

Key Concept

Relative Motion and Catch-Up Scenarios

Question 5Question →

A logistics company dispatches a delivery van traveling on a straight highway at a constant speed of $40$ mph. Exactly $90$ minutes later, a courier is dispatched from the same location to intercept the van, traveling on the same highway at a constant speed of $60$ mph. How many hours will it take the courier to catch up to the delivery van after departing?

4.5

0.6

3

1.8

Show answer & explanation

Answer: $3$

Answer

It will take

3

hours for the courier to catch up to the delivery van.

First, the van's head start of 90 minutes must be converted to 1.5 hours. Traveling at 40 mph for 1.5 hours puts the van 60 miles ahead of the starting point. Next, because the vehicles travel in the same direction, their relative speed is the difference between their individual speeds: 60 mph - 40 mph = 20 mph. This means the courier gains 20 miles on the van every hour. Finally, divide the 60-mile gap by the 20 mph relative speed to find the time: 60 / 20 = 3 hours.

Step-by-Step Solution

1

Convert the van's head start time from minutes to hours.

90 \text{ minutes} = 1.5 \text{ hours}

Speed is given in miles per hour (mph), so all time units must be in hours to maintain consistency.

2

Calculate the initial distance gap created by the van's head start.

40 \text{ mph} \times 1.5 \text{ hours} = 60 \text{ miles}

Distance equals rate multiplied by time. This is the gap the courier must close.

3

Determine the relative speed of the courier closing the gap.

60 \text{ mph} - 40 \text{ mph} = 20 \text{ mph}

Because both vehicles are moving in the same direction, subtract the van's speed from the courier's speed to find how fast the distance between them is shrinking.

4

Calculate the time required for the courier to close the initial gap.

60 \text{ miles} / 20 \text{ mph} = 3 \text{ hours}

Time equals distance divided by rate. Dividing the initial gap by the relative speed gives the catch-up time.

Key Concept

Relative motion in the same direction requires subtracting the speed of the slower object from the faster object to find the relative rate of closure.

Estimated Time:1m 30s

Question 6Question →

A passenger train leaves West Station heading east toward East Station, which is $270$ miles away, traveling at a constant speed of $60$ mph. Exactly $1$ hour later, a freight train leaves East Station heading west toward West Station on a parallel track at a constant speed of $45$ mph. How many hours after the passenger train's departure will the two trains pass each other?

Show answer & explanation

Answer: 3

Answer

3 hours

The passenger train travels 60 miles in its first hour, leaving 210 miles between the trains. Traveling toward each other, their combined relative speed is 105 mph. It takes them 2 hours to cover the 210 miles together. Adding the initial 1 hour, the total time since the passenger train departed is 3 hours.

Step-by-Step Solution

1

Calculate the distance covered by the passenger train during its 1-hour head start.

60 \text{ mph} \times 1 \text{ hour} = 60 \text{ miles}

This establishes how much distance is covered before the second train begins moving.

2

Find the remaining distance between the two trains.

270 \text{ miles} - 60 \text{ miles} = 210 \text{ miles}

This is the remaining gap the two trains must close together once both are moving.

3

Calculate the combined (relative) speed of the two trains.

60 \text{ mph} + 45 \text{ mph} = 105 \text{ mph}

Because they are traveling toward each other from opposite directions, their speeds add up to close the distance.

4

Determine the time it takes to close the remaining 210 miles.

210 \text{ miles} \div 105 \text{ mph} = 2 \text{ hours}

Using the formula Time = Distance / Rate.

5

Calculate the total time since the passenger train's departure.

1 \text{ hour} + 2 \text{ hours} = 3 \text{ hours}

The question specifically asks for the time after the passenger train's departure, which includes its initial 1-hour head start.

Key Concept

Relative motion and converging objects with staggered start times

Question 7Question →

A military transport convoy departs from a base and travels on a straight route at a constant speed of $30$ mph. Exactly $15$ minutes later, a tactical vehicle is dispatched along the same route to deliver updated orders, traveling at a constant speed of $45$ mph. How many minutes will it take for the tactical vehicle to catch up to the convoy from the time the tactical vehicle departs?

6

60

30

300

Show answer & explanation

Answer: 30

Answer

30

The correct answer accurately calculates the

7.5

-mile head start distance (

30 \text{ mph} \times 0.25 \text{ hours}

) and divides it by the

15

mph relative speed (

45 \text{ mph} - 30 \text{ mph}

) to find a catch-up time of

0.5

hours, which equals

30

minutes.

Step-by-Step Solution

1

Convert the convoy's head start time from minutes to hours.

15 \text{ minutes} = 0.25 \text{ hours}

.

Vehicle speeds are given in miles per hour (mph), so the time must be converted to hours for unit consistency.

2

Calculate the distance the convoy traveled during its head start.

30 \text{ mph} \times 0.25 \text{ hours} = 7.5 \text{ miles}

.

This establishes the initial distance gap that the tactical vehicle needs to close.

3

Determine the relative catch-up speed of the tactical vehicle.

45 \text{ mph} - 30 \text{ mph} = 15 \text{ mph}

.

Because the vehicles are traveling in the same direction, you subtract the slower speed from the faster speed to find the rate at which the gap is closing.

4

Calculate the time required to close the gap in hours.

7.5 \text{ miles} \div 15 \text{ mph} = 0.5 \text{ hours}

.

Time equals distance divided by rate (

t = \frac{d}{r}

).

5

Convert the catch-up time back to minutes.

0.5 \text{ hours} \times 60 \text{ minutes/hour} = 30 \text{ minutes}

.

The question specifically asks for the time in minutes.

Key Concept

Catch-up scenarios require calculating the initial distance gap and dividing it by the relative speed (the difference in speeds) between the two moving objects.

Question 8Question →

A scout rover leaves a forward operating base and travels along a straight path at a constant speed of $40$ mph. Exactly $30$ minutes later, a heavily armored command vehicle leaves the same base along the same path at a constant speed of $55$ mph to overtake the rover. How many minutes will the command vehicle need to drive before it catches up to the scout rover?

Show answer & explanation

Answer: 80; 80 minutes; 80 mins; eighty

Answer

80

The scout rover establishes a 20-mile lead during its 30-minute (0.5 hour) head start. The command vehicle closes this 20-mile gap at a relative speed of 15 mph (55 mph - 40 mph). It takes 4/3 hours to close the gap, which converts to exactly 80 minutes.

Step-by-Step Solution

1

Convert the rover's head start time from minutes to hours.

30

minutes is equal to

0.5

hours.

Speed is given in miles per hour (mph), so time must be in hours to accurately calculate the distance.

2

Calculate the head start distance of the scout rover.

40 \text{ mph} \times 0.5 \text{ hours} = 20 \text{ miles}

.

This establishes the initial gap that the command vehicle must close.

3

Calculate the relative catch-up speed.

55 \text{ mph} - 40 \text{ mph} = 15 \text{ mph}

.

Since both vehicles are traveling in the same direction, subtract the slower speed from the faster speed to find how fast the gap is closing.

4

Calculate the time required to catch up in hours.

20 \text{ miles} \div 15 \text{ mph} = \frac{4}{3} \text{ hours}

.

Time equals distance divided by rate.

5

Convert the catch-up time back to minutes.

\frac{4}{3} \times 60 = 80 \text{ minutes}

.

The question specifically asks for the driving time in minutes.

Key Concept

Relative Motion and Catch-Up Scenarios

Question 9Question →

Two rescue helicopters, operating in the same coastal region, are $360$ miles apart. They take off at the same time and fly directly toward each other. Helicopter A flies at a constant speed of $110$ mph, and Helicopter B flies at a constant speed of $130$ mph. How many minutes will it take for the two helicopters to meet?

1080

180

90

110

Show answer & explanation

Answer: 90

Answer

It will take 90 minutes for the two helicopters to meet.

The correct answer is 90. When two objects travel directly toward each other, their relative rate is the sum of their individual speeds. The combined speed is

110 + 130 = 240

mph. The time to meet is the total distance divided by the relative speed:

360 / 240 = 1.5

hours. Finally, to convert hours to minutes, multiply by 60:

1.5 \times 60 = 90

minutes.

Step-by-Step Solution

1

Calculate the combined relative speed of the two helicopters.

110 + 130 = 240

mph

When two objects are moving directly toward each other, their relative speed is the sum of their individual speeds.

2

Calculate the time in hours it takes for them to meet.

360 / 240 = 1.5

hours

Time is equal to the total distance divided by the relative speed.

3

Convert the time from hours to minutes.

1.5 \times 60 = 90

minutes

The question specifically asks for the time in minutes.

Key Concept

Relative Motion and Time Conversion

Question 10Question →

During a naval training exercise, a submarine travels submerged in a straight line at a constant speed of $15$ knots. Exactly $2$ hours later, a destroyer departs from the same location and pursues the submarine along the exact same path at a constant speed of $25$ knots. How many hours will it take for the destroyer to overtake the submarine from the time the destroyer begins moving?

Show answer & explanation

Answer: 3

Answer

It will take 3 hours for the destroyer to catch up to the submarine.

To solve a catch-up scenario, first determine the initial distance between the two objects. The submarine travels at

15

knots for

2

hours, creating a

30

nautical mile gap. Next, find the relative speed by subtracting the slower speed from the faster speed (

25 - 15 = 10

knots) because they are moving in the same direction. Finally, divide the initial gap by the relative speed (

30 \div 10 = 3

) to find the time it takes the destroyer to close the gap.

Step-by-Step Solution

1

Calculate the head-start distance of the submarine.

30

nautical miles

The submarine travels at

15

knots for

2

hours before the destroyer starts.

2

Calculate the relative catch-up speed.

10

knots

The destroyer travels in the same direction, so the relative speed is the difference between their speeds (

25 - 15

).

3

Calculate the time required to close the distance gap.

3

hours

Time equals distance divided by relative rate. Dividing the

30

nautical mile gap by the relative speed of

10

knots gives

3

hours.

Key Concept

Relative Motion and Catch-Up Scenarios

Question 11Question →

A naval supply ship leaves port heading due east at a constant speed of $24$ knots. Exactly $1$ hour and $45$ minutes later, a destroyer leaves the same port to intercept the supply ship, traveling on the exact same course at a constant speed of $36$ knots. How many hours will it take for the destroyer to catch up with the supply ship, measured from the moment the destroyer leaves port?

Show answer & explanation

Answer: 3.5; 3.5 hours; 3 1/2; 3 1/2 hours; 3.5 hrs; 7/2

Answer

3.5

The destroyer must close an initial gap of

42

nautical miles. Since it travels

12

knots faster than the supply ship, it closes the gap at a relative rate of

12

knots, which takes exactly

3.5

hours.

Step-by-Step Solution

1

Convert the head start time from hours and minutes into a decimal or fraction of hours.

1

hour and

45

minutes =

1.75

hours (or

\frac{7}{4}

hours).

Time units must properly match the speed units (knots, which are nautical miles per hour) before calculating distance.

2

Calculate the distance the supply ship traveled during its head start.

24 \text{ knots} \times 1.75 \text{ hours} = 42 \text{ nautical miles}

.

This establishes the initial gap distance the destroyer needs to close.

3

Determine the relative speed at which the destroyer is closing the gap.

36 \text{ knots} - 24 \text{ knots} = 12 \text{ knots}

.

Because both ships are traveling in the same direction, you subtract the slower speed from the faster speed to find the catch-up rate.

4

Calculate the time required to close the gap.

\frac{42 \text{ nautical miles}}{12 \text{ knots}} = 3.5 \text{ hours}

.

Time equals distance divided by rate. The gap distance divided by the relative speed gives the exact time to intercept.

Key Concept

Relative Motion and Catch-Up Scenarios

Question 12Question →

A commercial cargo flight leaves an airport at 06:00 AM, heading due east at a constant cruising speed of $300$ mph. At 07:30 AM, a passenger jet leaves the same airport, flying along the exact same route at a constant cruising speed of $600$ mph. At what time will the passenger jet catch up to the cargo flight?

08:00 AM

08:30 AM

09:00 AM

09:20 AM

Show answer & explanation

Answer: 09:00 AM

Answer

09:00 AM

The cargo flight has a

1.5

-hour head start, covering

450

miles (

300 \text{ mph} \times 1.5 \text{ hours}

). Because both aircraft are traveling in the same direction, the passenger jet closes this gap at a relative speed of

300

mph (

600 \text{ mph} - 300 \text{ mph}

). It takes the passenger jet

1.5

hours to close the

450

-mile gap (

450 \div 300 = 1.5

). Adding

1.5

hours (

1

hour and

30

minutes) to the passenger jet's departure time of 07:30 AM gives a final catch-up time of 09:00 AM.

Step-by-Step Solution

1

Calculate the time difference between the departures.

1.5

hours.

From 06:00 AM to 07:30 AM is exactly

1

hour and

30

minutes, or

1.5

hours. This determines how long the cargo flight traveled before the passenger jet departed.

2

Calculate the head-start distance of the cargo flight.

450

miles.

Using Distance = Rate

\times

Time, multiply the cargo flight's speed by its head-start time:

300 \text{ mph} \times 1.5 \text{ hours} = 450 \text{ miles}

.

3

Determine the relative speed of the two aircraft.

300

mph.

Because both aircraft are traveling in the same direction, subtract the slower speed from the faster speed to find how quickly the gap is closing:

600 \text{ mph} - 300 \text{ mph} = 300 \text{ mph}

.

4

Calculate the time required for the passenger jet to catch up.

1.5

hours.

Divide the head-start distance by the relative speed:

450 \text{ miles} \div 300 \text{ mph} = 1.5 \text{ hours}

.

5

Add the catch-up time to the passenger jet's departure time.

09:00 AM.

1.5

hours is

1

hour and

30

minutes. Adding this to 07:30 AM gives 09:00 AM.

Key Concept

Relative Motion (Same Direction)

Relative Motion and Catch-Up Scenarios

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