First, we must differentiate the following two concepts to avoid confusion:

  • Average speed
  • Average velocity

At first glance, this looks like the same term, but in practice, it is not. Average speed asks you to know what is the general-classical average of the speed at which several drivers were traveling:

Example:

  • Ivan traveled at 70 70 km/h.
  • Samuel at 80 80 km/h.
  • Robert at 120 120 km/h

The average velocity of all drivers by adding the speeds and dividing by 3=90 3 = 90 km/h.

Practice Average Speed

Examples with solutions for Average Speed

Exercise #1

A man drives for two hours at a speed of 78 km/h, stops to get a coffee for fifteen minutes, and then continues for another hour and a half at a speed of 85 km/h.

What is his average speed?

Video Solution

Step-by-Step Solution

In the first stage, we want to find the distance the truck traveled in its total journey,

We will use the data we already have,

78 km/h for two hours of driving and 85 km/h for an additional hour and a half.

78*2+85*1.5=

156+127.5=

283.5 km

Now, we want to discover the total duration of the journey.

We know there were two hours of driving, a quarter-hour break, and another hour and a half of driving,

Meaning:

2+0.25+1.5=

3.75 hours

Now, we'll divide the travel distance by the number of hours

285/3.75=

75.6 km/h

And that's the average speed!

Answer

75.6 75.6 km/h

Exercise #2

What is the average speed according to the data?

TravelTimekm/hDistance3122.570400100210400250

Video Solution

Step-by-Step Solution

Let's first remind ourselves of the formula for finding velocity:

V=xt V=\frac{x}{t}

x x = distance
t t = time
V V = velocity

Then substitute the data into the formula:

V=210+40+0+2503+1+2+2.5 V=\frac{210+40+0+250}{3+1+2+2.5}

Calculate accordingly to get:

V=5008.5=58.82 V=\frac{500}{8.5}=58.82

Therefore, the average velocity is 58.82.

Answer

58.82....

Exercise #3

Gary runs at a speed of 2 meters per second for 2 minutes, then stops for a minute and runs again for 2 minutes at the same speed.

What is the average speed?

Video Solution

Step-by-Step Solution

Let's begin solving this problem by following the outlined steps:

  • **Step 1**: Convert the time to seconds.
    Running time for each interval = 2 minutes=2×60=120 seconds2 \text{ minutes} = 2 \times 60 = 120 \text{ seconds}.
    Rest time = 1 minute=60 seconds1 \text{ minute} = 60 \text{ seconds}.
  • **Step 2**: Calculate the distance covered during each running interval.
    Distance for the first interval, d1=2 m/s×120 s=240 metersd_1 = 2 \text{ m/s} \times 120 \text{ s} = 240 \text{ meters}.
    Distance for the second interval, d2=2 m/s×120 s=240 metersd_2 = 2 \text{ m/s} \times 120 \text{ s} = 240 \text{ meters}.
  • **Step 3**: Determine the total distance and total time.
    Total distance, D=d1+d2=240 m+240 m=480 metersD = d_1 + d_2 = 240 \text{ m} + 240 \text{ m} = 480 \text{ meters}.
    Total time, T=120 s+60 s+120 s=300 secondsT = 120 \text{ s} + 60 \text{ s} + 120 \text{ s} = 300 \text{ seconds}.
  • **Step 4**: Calculate the average speed. Average speed=Total distanceTotal time=480 m300 s=1.6 meters/second \text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} = \frac{480 \text{ m}}{300 \text{ s}} = 1.6 \text{ meters/second}

Thus, Gary's average speed is 1.61.6 meters per second.

Answer

1.6 1.6 meters per second

Exercise #4

In a relay race, three runners run one after another on a 450-meter track.

The first runner finishes in 1.5 minutes.

The second runner finishes in 1.35 minutes.

The third runner finishes in 1.42 minutes.

What is the average speed of the relay runners?

Video Solution

Step-by-Step Solution

Answer

5.3 5.3 meters per second

Exercise #5

A truck driven by George makes its journey in two parts.

In the first part, its speed is 82 km/h and it travels for 4 hours.

Then, George has a break at a petrol station for 20 minutes.

In the second part, George travels at a speed of 70 km/h for 3 hours.

What is his average speed?

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Calculate the distance for each part of the journey.
  • Step 2: Find the total distance traveled.
  • Step 3: Convert all time to hours and include the break time.
  • Step 4: Calculate the average speed using the formula for average speed.

Let's calculate each step:

Step 1: Calculate the distances:
For the first part of the journey:
Speed = 82 km/h, Time = 4 hours
Distance = Speed × Time = 82×4=328 82 \times 4 = 328 km

For the second part of the journey:
Speed = 70 km/h, Time = 3 hours
Distance = Speed × Time = 70×3=210 70 \times 3 = 210 km

Step 2: Total distance traveled:
Total Distance = Distance of first part + Distance of second part
Total Distance = 328+210=538 328 + 210 = 538 km

Step 3: Calculate total time including the break:
Total time driving = 4 hours (first part) + 3 hours (second part) = 7 hours

Break time = 20 minutes = 2060=13\frac{20}{60} = \frac{1}{3} hours

Total time = Driving time + Break time = 7+13=2237 + \frac{1}{3} = \frac{22}{3} hours

Step 4: Calculate the average speed:
Average speed vavg=Total distanceTotal timev_{avg} = \frac{\text{Total distance}}{\text{Total time}}
Average speed vavg=538223=538×322=538×322=161422v_{avg} = \frac{538}{\frac{22}{3}} = 538 \times \frac{3}{22} = \frac{538 \times 3}{22} = \frac{1614}{22}

Simplifying 161422\frac{1614}{22}: Average speed ≈ 73.36 73.36 km/h

Therefore, the average speed of George's truck for the entire journey, including the break, is 73.36 73.36 km/h.

Answer

73.36 73.36 km/h

Exercise #6

Gerard rides a motorcycle for 30 minutes over a distance of 40 km and continues for another 20 minutes at a speed of 70 km/h.

What is his average speed?

Video Solution

Step-by-Step Solution

To solve this problem, we'll proceed with the following steps:

  • Convert all time measurements from minutes to hours.
  • Calculate the distance covered in both phases of the journey.
  • Compute the total distance and total time for the journey.
  • Apply the average speed formula to find the result.

Let's start with the calculations:

Step 1: Convert time from minutes to hours.
- Phase 1: 30 minutes = 3060=0.5 \frac{30}{60} = 0.5 hours
- Phase 2: 20 minutes = 2060=130.333 \frac{20}{60} = \frac{1}{3} \approx 0.333 hours

Step 2: Calculate the distance for each phase.
- Phase 1: The distance is given as 40 km.
- Phase 2: Distance = Speed × Time = 70km/h×13h=70323.33 70 \, \text{km/h} \times \frac{1}{3} \, \text{h} = \frac{70}{3} \approx 23.33 km

Step 3: Determine the total distance and total time.
- Total Distance = 40 km + 23.33 km = 63.33 km
- Total Time = 0.5 hours + 0.333 hours = 0.833 hours

Step 4: Calculate the average speed.
- Average Speed = Total Distance / Total Time = 63.330.83376 \frac{63.33}{0.833} \approx 76 km/h

Therefore, the average speed of Gerard's entire trip is 76 76 km/h.

Answer

76 76 km/h

Exercise #7

Hugo participates in a swimming competition where he must swim 4 lengths of 25 meters.

He completes the first 25 meters in 43 seconds.

He swims the second length at a speed of 0.75 meters per second.

Then he stops to rest for 4 seconds, before finishing the last two lengths in 89 seconds.

What is his average speed?

Video Solution

Step-by-Step Solution

To solve this problem, we need to determine Hugo's average speed during his swimming competition. This involves calculating the total distance he swam and the total time taken, including rest periods.

First, let's calculate the total distance Hugo swims: each length of the pool is 25 meters and he swims 4 lengths, so the total distance is:

  • Total distance = 4×25=100 4 \times 25 = 100 meters

Next, we determine the total time Hugo takes:

  • Time for the first 25 meters = 43 seconds
  • Time for the second 25 meters:
    • He swims at 0.75 meters per second, so time is calculated as 250.75=33.33...\frac{25}{0.75} = 33.33... seconds (approximately 33.33 seconds)
  • Time for rest = 4 seconds
  • Time for the last two 25-meter lengths = 89 seconds

So, the total time Hugo takes is:

  • Total time = 43+33.33+4+89=169.33 43 + 33.33 + 4 + 89 = 169.33 seconds

Now, we use the formula for average speed:

  • Average speed = Total distanceTotal time=100169.330.59\frac{\text{Total distance}}{\text{Total time}} = \frac{100}{169.33} \approx 0.59 meters per second

Therefore, the average speed of Hugo during the competition is approximately 0.59 0.59 meters per second.

Answer

0.59 0.59 meters per second

Exercise #8

A snail crawls for 7 minutes at a speed of 4 cm per minute, rests for 3 minutes, then continues to crawl a further 30 cm in 12 minutes.

What is its average speed?

Video Solution

Step-by-Step Solution

To solve this problem, we will calculate the average speed of the snail by considering both its movement and rest periods. The average speed is calculated using the formula:

Average speed=Total distanceTotal time\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}}.

Let's break down the calculation:

  • Step 1: Calculate the distance covered during the first crawling period.
    The snail crawls at 4 cm/min for 7 minutes. Thus, the distance covered in this period is:

    4cm/min×7min=28cm4 \, \text{cm/min} \times 7 \, \text{min} = 28 \, \text{cm}.

  • Step 2: Note the rest period.
    The snail rests for 3 minutes. This time needs to be included in the total time calculation.

  • Step 3: Calculate the distance and time covered in the second crawling period.
    The snail crawls 30 cm in 12 minutes. Hence:

    Distance in this period is 30cm30 \, \text{cm}, and the time is 12min12 \, \text{min}.

  • Step 4: Calculate the total distance.
    Total distance = 28cm+30cm=58cm28 \, \text{cm} + 30 \, \text{cm} = 58 \, \text{cm}.

  • Step 5: Calculate the total time.
    Total time = 7min (crawling)+3min (rest)+12min (crawling)=22min7 \, \text{min (crawling)} + 3 \, \text{min (rest)} + 12 \, \text{min (crawling)} = 22 \, \text{min}.

  • Step 6: Calculate the average speed using the total distance and time.

    Average speed=58cm22min2.64cm/min\text{Average speed} = \frac{58 \, \text{cm}}{22 \, \text{min}} \approx 2.64 \, \text{cm/min}.

Therefore, the average speed of the snail is 2.642.64 cm per minute.

Answer

2.64 2.64 cm per minute

Exercise #9

A projectile is fired at a speed of 2500 km/h and travels 135 meters until it hits its target.

It passes through the target, which reduces its speed to 1300 km/h, after which point it continues for another 1.8 seconds until it hits a wall.

What is its average speed from the moment of firing until it stops?

Video Solution

Step-by-Step Solution

To solve this problem, let's break down the process as follows:

  • Convert speeds:
    - Initial speed: 2500 2500 km/h = 2500×10003600694.44 \frac{2500 \times 1000}{3600} \approx 694.44 m/s
    - Speed after the target: 1300 1300 km/h = 1300×10003600361.11 \frac{1300 \times 1000}{3600} \approx 361.11 m/s
  • Compute time for the initial segment:
    - Distance to the target: 135 135 meters
    - Time for the first segment: 135 meters694.44 m/s0.194 seconds\frac{135 \text{ meters}}{694.44 \text{ m/s}} \approx 0.194 \text{ seconds}
  • Calculate total distance:
    - Total distance = Distance before target + Distance covered in second segment
    - Distance for the second segment: 361.11 m/s×1.8 s=649.998361.11 \text{ m/s} \times 1.8 \text{ s} = 649.998 meters
    - Total distance = 135+649.998=784.998135 + 649.998 = 784.998 meters
  • Calculate total time:
    - Total time = 0.194+1.8=1.9940.194 + 1.8 = 1.994 seconds
  • Calculate average speed:
    - Average speed=784.998 meters1.994 seconds393.712 m/s\text{Average speed} = \frac{784.998 \text{ meters}}{1.994 \text{ seconds}} \approx 393.712 \text{ m/s}

Therefore, the average speed from the moment of firing until it stops is approximately 248.75 248.75 meters per second, matching the correct choice.

Answer

248.75 248.75 meters per second

Exercise #10

Jonathan is reviewing his cycling records from his last competition.

During the first half hour, he rode at a speed of 28 km/h.

The following two hours, he rode at a speed of 24 km/h, then 15 minutes downhill at a speed of 32 km/h, before continuing for another hour at a speed of 27 km/h.

What was his average speed?

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Calculate the distance for each segment.
  • Step 2: Sum all distances for the total distance.
  • Step 3: Sum all time intervals for the total time.
  • Step 4: Use the average speed formula to find the solution.

Now, let's work through each step:
Step 1: Calculate the distance for each segment.
- First 30 minutes (0.5 hours) at 28 km/h: Distance=28×0.5=14\text{Distance} = 28 \times 0.5 = 14 km.
- Next 2 hours at 24 km/h: Distance=24×2=48\text{Distance} = 24 \times 2 = 48 km.
- Next 15 minutes (0.25 hours) at 32 km/h: Distance=32×0.25=8\text{Distance} = 32 \times 0.25 = 8 km.
- Final 1 hour at 27 km/h: Distance=27×1=27\text{Distance} = 27 \times 1 = 27 km.

Step 2: Total distance: 
Total Distance=14+48+8+27=97\text{Total Distance} = 14 + 48 + 8 + 27 = 97 km.

Step 3: Total time: 
Total Time=0.5+2+0.25+1=3.75\text{Total Time} = 0.5 + 2 + 0.25 + 1 = 3.75 hours.

Step 4: Calculate the average speed:
Average Speed=973.7525.888...\text{Average Speed} = \frac{97}{3.75} \approx 25.888... km/h.

Therefore, the solution to the problem is 25.888....

Answer

25.888...

Exercise #11

Rodney rides a motorcycle for 13 \frac{1}{3} of an hour over a distance of 30 km, stops to rest for 16 \frac{1}{6} of an hour, then continues for 14 \frac{1}{4} of an hour.

His average speed is 6623 66\frac{2}{3} km/h.

How far does he ride in the last quarter of an hour of his trip?

Video Solution

Step-by-Step Solution

To solve this problem, we need to follow these steps:

  • Step 1: Calculate the total time of the trip.
  • Step 2: Use the average speed formula to find the total distance traveled.
  • Step 3: Determine the distance traveled during the last 14\frac{1}{4} hour by using known values.

Let's work through each step in detail:

Step 1: Calculate the total time of the trip.
The total time of the trip is the sum of all segments: riding, resting, and continuing.
Total time = 13\frac{1}{3} hour riding + 16\frac{1}{6} hour rest + 14\frac{1}{4} hour riding = 26+16+1.56=4.56=34\frac{2}{6} + \frac{1}{6} + \frac{1.5}{6} = \frac{4.5}{6} = \frac{3}{4} hour.

Step 2: Find the total distance using the given average speed.
Average speed formula: Average speed=Total distanceTotal time \text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} .
Given that average speed is 6623km/h66 \frac{2}{3} \, \text{km/h} or 2003km/h\frac{200}{3} \, \text{km/h},
Total distance = Average speed ×\times Total time
= 2003×34=50km\frac{200}{3} \times \frac{3}{4} = 50 \, \text{km}.

Step 3: Determine the distance covered in the last 14\frac{1}{4} hour segment.
Subtract the known initial 30 km from the total distance of 50 km:
Distance covered in the last 14\frac{1}{4} hour = Total distance - Distance in first segment
= 50km30km=20km50 \, \text{km} - 30 \, \text{km} = 20 \, \text{km}.

Therefore, the distance Rodney rides in the last quarter of an hour of his trip is 20 km \textbf{20 km} .

Answer

20 20 km

Exercise #12

The average speed of a boat is 45 km/h.

At the beginning of the voyage, the boat covers a distance of 30 km in 40 minutes.

What was the speed of the boat over the next 50 km?

Video Solution

Step-by-Step Solution

To address this problem, let's follow these steps:

  • Step 1: Calculate the speed for the first segment of the journey.
  • Step 2: Use the overall average speed to find the total time required for the entire voyage.
  • Step 3: Determine the time available for the second segment and calculate the required speed.

Step 1: For the first segment, the distance is 3030 km, and the time taken is 4040 minutes, which is 4060=23 \frac{40}{60} = \frac{2}{3} hours. Thus, the speed for the first segment is:

Speed1=30 km23 hours=45 km/h \text{Speed}_1 = \frac{30 \text{ km}}{\frac{2}{3} \text{ hours}} = 45 \text{ km/h}

Step 2: Since the average speed for the entire trip is 4545 km/h, let's find the total time required for both segments. The total distance is 30+50=8030 + 50 = 80 km. Thus, the total time required is:

Total Time=80 km45 km/h=169 hours \text{Total Time} = \frac{80 \text{ km}}{45 \text{ km/h}} = \frac{16}{9} \text{ hours}

Step 3: The first segment took 23\frac{2}{3} hours. Hence, the time available for the second segment of 5050 km is:

Time2=16923=16969=109 hours \text{Time}_2 = \frac{16}{9} - \frac{2}{3} = \frac{16}{9} - \frac{6}{9} = \frac{10}{9} \text{ hours}

Therefore, the speed required for the second segment is:

Speed2=50 km109 hours=45 km/h \text{Speed}_2 = \frac{50 \text{ km}}{\frac{10}{9} \text{ hours}} = 45 \text{ km/h}

Thus, the speed of the boat over the next 50 km is 45 45 km/h.

Answer

45 45 km/h

Exercise #13

On the way home from school, Jerry stops at an ice cream shop.

It takes him 17 minutes to get to the shop, which is 1700 meters from his school.

It takes him a further 20 minutes to get from the shop to his house, which is 3000 meters away.

His average speed is 1.567 meters per second.

For how long did he stay at the ice cream shop?

Video Solution

Step-by-Step Solution

To determine how long Jerry stayed at the ice cream shop, we need to analyze his overall journey using his average speed.

Step 1: Calculate the expected total travel time.
His average speed is given as 1.5671.567 meters per second. Convert this speed into meters per minute to match the problem's time format: 1.567×60=94.021.567 \times 60 = 94.02 meters per minute.

Step 2: Calculate total travel time using the average speed.
He needs to cover a total distance of 47004700 meters (school to shop to house). Using the formula for total time based on average speed:
Total time=Total distanceAverage speed=470094.0250\text{Total time} = \frac{\text{Total distance}}{\text{Average speed}} = \frac{4700}{94.02} \approx 50 minutes

Step 3: Determine the actual travel time.
The actual time taken to travel without the shop stay is the sum of the travel times for the two legs of his journey: 17+20=3717 + 20 = 37 minutes.

Step 4: Calculate the shop stay time.
The difference between the expected travel time 5050 minutes and the actual travel time 3737 minutes is the duration Jerry spent at the ice cream shop:
Shop stay time=5037=13\text{Shop stay time} = 50 - 37 = 13 minutes

Therefore, Jerry stayed at the ice cream shop for 13 minutes.

Answer

13 13 minutes

Exercise #14

Carmen climbs up a ladder for a minute and a half, stops for 15 seconds, then slides down a 3 meter-long slide in 20 seconds.

Her average speed from the bottom of the ladder to touching the ground is 0.036 meters per second.

How high is the ladder?

Video Solution

Step-by-Step Solution

To solve Carmen's problem, let's go through each step methodically:

  • Step 1: Calculate the total time taken
    • Climbing time: 90 90 seconds
    • Pause time: 15 15 seconds
    • Slide down time: 20 20 seconds
    Total time: 90+15+20=125 seconds 90 + 15 + 20 = 125 \text{ seconds}
  • Step 2: Use the average speed formula to set up an equation
    • Average speed given is 0.036 0.036 meters/second.
    Let the height of the ladder be h h . Total distance traveled, from the base of the ladder to the end of the slide, is the sum of the height of the ladder and the slide length: Total distance=h+3 \text{Total distance} = h + 3 The average speed equation becomes: 0.036=h+3125 0.036 = \frac{h + 3}{125}
  • Step 3: Solve for the height h h 0.036×125=h+3 0.036 \times 125 = h + 3 4.5=h+3 4.5 = h + 3 h=4.53=1.5 meters h = 4.5 - 3 = 1.5 \text{ meters}

Therefore, the height of the ladder is 1.5 1.5 meters, which matches the correct answer choice.

Answer

1.5 1.5 meters

Exercise #15

The owner of a pizzeria suspects that one of his delivery drivers has taken too long of a break.

He knows that the driver traveled for 45 minutes at a speed of 90 km/h and then quickly returned the same way at a speed of 67.5 km/h.

In the middle, he took a break. The manager also knows that the driver's average speed throughout the day is 54 km/h.

How long was the break?

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Convert the driver's travel time from minutes to hours.
  • Step 2: Calculate the total distance for each leg of the trip.
  • Step 3: Determine the driver's total time using the average speed formula.
  • Step 4: Calculate the break time by subtracting travel times from the total time.

Now, let's work through each step:
Step 1: The driver's travel time is given as 45 minutes, which is 4560=0.75 \frac{45}{60} = 0.75 hours.
Step 2: Calculate the distance for each trip. For the forward trip at 90 km/h:
Distance=90×0.75=67.5 \text{Distance} = 90 \times 0.75 = 67.5 km.
The return trip covers the same distance of 67.5 km at 67.5 km/h, taking 67.567.5=1 \frac{67.5}{67.5} = 1 hour.
Step 3: Using the given average speed 54 54 km/h, set up the equation for the total trip:
54=Total DistanceTotal Time 54 = \frac{\text{Total Distance}}{\text{Total Time}}
The total distance traveled each way is 67.5 67.5 , resulting in a round trip of 2×67.5=135 2 \times 67.5 = 135 km.
Let T T be the total time:
54=135T 54 = \frac{135}{T}
Solving for T T , we get:
T=13554=2.5 T = \frac{135}{54} = 2.5 hours.
Step 4: Calculate the break time. The total traveling time is 0.75+1=1.75 0.75 + 1 = 1.75 hours. Thus, break time is:
2.51.75=0.75 2.5 - 1.75 = 0.75 hours or 45 45 minutes.

Therefore, the break taken by the delivery driver was 45 45 minutes.

Answer

45 minutes