Average Speed - Examples, Exercises and Solutions

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!

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

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

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

Then substitute the data into the formula:

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

Calculate accordingly to get:

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

Therefore, the average velocity is 58.82.

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

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

Thus, Gary's average speed is $1.6$ meters per second.

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 = $82 \times 4 = 328$ km

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

Step 2: Total distance traveled:
Total Distance = Distance of first part + Distance of second part
Total Distance = $328 + 210 = 538$ 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 = $\frac{20}{60} = \frac{1}{3}$ hours

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

Step 4: Calculate the average speed:
Average speed $v_{avg} = \frac{\text{Total distance}}{\text{Total time}}$
Average speed $v_{avg} = \frac{538}{\frac{22}{3}} = 538 \times \frac{3}{22} = \frac{538 \times 3}{22} = \frac{1614}{22}$

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

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

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 = $\frac{30}{60} = 0.5$ hours
- Phase 2: 20 minutes = $\frac{20}{60} = \frac{1}{3} \approx 0.333$ hours

Step 2: Calculate the distance for each phase.
- Phase 1: The distance is given as 40 km.
- Phase 2: Distance = Speed × Time = $70 \, \text{km/h} \times \frac{1}{3} \, \text{h} = \frac{70}{3} \approx 23.33$ 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 = $\frac{63.33}{0.833} \approx 76$ km/h

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

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 = $4 \times 25 = 100$ 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 $\frac{25}{0.75} = 33.33...$ 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 = $43 + 33.33 + 4 + 89 = 169.33$ seconds

Now, we use the formula for average speed:

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

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

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:

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

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:

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

• 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 $30 \, \text{cm}$, and the time is $12 \, \text{min}$.

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

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

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

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

Therefore, the average speed of the snail is $2.64$ cm per minute.

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

• Convert speeds:
  - Initial speed: $2500$ km/h = $\frac{2500 \times 1000}{3600} \approx 694.44$ m/s
  - Speed after the target: $1300$ km/h = $\frac{1300 \times 1000}{3600} \approx 361.11$ m/s
• Compute time for the initial segment:
  - Distance to the target: $135$ meters
  - Time for the first segment: $\frac{135 \text{ meters}}{694.44 \text{ m/s}} \approx 0.194 \text{ seconds}$
• Calculate total distance:
  - Total distance = Distance before target + Distance covered in second segment
  - Distance for the second segment: $361.11 \text{ m/s} \times 1.8 \text{ s} = 649.998$ meters
  - Total distance = $135 + 649.998 = 784.998$ meters
• Calculate total time:
  - Total time = $0.194 + 1.8 = 1.994$ seconds
• Calculate average speed:
  - $\text{Average speed} = \frac{784.998 \text{ meters}}{1.994 \text{ seconds}} \approx 393.712 \text{ m/s}$

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

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: $\text{Distance} = 28 \times 0.5 = 14$ km.
- Next 2 hours at 24 km/h: $\text{Distance} = 24 \times 2 = 48$ km.
- Next 15 minutes (0.25 hours) at 32 km/h: $\text{Distance} = 32 \times 0.25 = 8$ km.
- Final 1 hour at 27 km/h: $\text{Distance} = 27 \times 1 = 27$ km.

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

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

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

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

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 $\frac{1}{4}$ 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 = $\frac{1}{3}$ hour riding + $\frac{1}{6}$ hour rest + $\frac{1}{4}$ hour riding = $\frac{2}{6} + \frac{1}{6} + \frac{1.5}{6} = \frac{4.5}{6} = \frac{3}{4}$ hour.

Step 2: Find the total distance using the given average speed.
Average speed formula: $\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}}$.
Given that average speed is $66 \frac{2}{3} \, \text{km/h}$ or $\frac{200}{3} \, \text{km/h}$,
Total distance = Average speed $\times$ Total time
= $\frac{200}{3} \times \frac{3}{4} = 50 \, \text{km}$.

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

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

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 $30$ km, and the time taken is $40$ minutes, which is $\frac{40}{60} = \frac{2}{3}$ hours. Thus, the speed for the first segment is:

Step 2: Since the average speed for the entire trip is $45$ km/h, let's find the total time required for both segments. The total distance is $30 + 50 = 80$ km. Thus, the total time required is:

Step 3: The first segment took $\frac{2}{3}$ hours. Hence, the time available for the second segment of $50$ km is:

Therefore, the speed required for the second segment is:

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

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 $1.567$ meters per second. Convert this speed into meters per minute to match the problem's time format: $1.567 \times 60 = 94.02$ meters per minute.

Step 2: Calculate total travel time using the average speed.
He needs to cover a total distance of $4700$ meters (school to shop to house). Using the formula for total time based on average speed:
$\text{Total time} = \frac{\text{Total distance}}{\text{Average speed}} = \frac{4700}{94.02} \approx 50$ 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: $17 + 20 = 37$ minutes.

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

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

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

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

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

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 $\frac{45}{60} = 0.75$ hours.
Step 2: Calculate the distance for each trip. For the forward trip at 90 km/h:
$\text{Distance} = 90 \times 0.75 = 67.5$ km.
The return trip covers the same distance of 67.5 km at 67.5 km/h, taking $\frac{67.5}{67.5} = 1$ hour.
Step 3: Using the given average speed $54$ km/h, set up the equation for the total trip:
$54 = \frac{\text{Total Distance}}{\text{Total Time}}$
The total distance traveled each way is $67.5$, resulting in a round trip of $2 \times 67.5 = 135$ km.
Let $T$ be the total time:
$54 = \frac{135}{T}$
Solving for $T$, we get:
$T = \frac{135}{54} = 2.5$ hours.
Step 4: Calculate the break time. The total traveling time is $0.75 + 1 = 1.75$ hours. Thus, break time is:
$2.5 - 1.75 = 0.75$ hours or $45$ minutes.

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