In a race, there are 4 members in a team. Each member has to cover 5 km one after another. If the total time taken is 30 minutes, then what would have been the average speed?
40\(\frac{{km}}{{hr}}\)
Calculating Average Speed in a Race
This question asks us to find the average speed of a team in a race. To find the average speed, we need to know the total distance covered and the total time taken. The formula for average speed is:
$$ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} $$
Step 1: Determine the Total Distance
The race involves a team of 4 members. Each member has to cover a distance of 5 km. Since they cover the distance one after another, the total distance covered by the entire team is the sum of the distances covered by each member.
- Number of members = 4
- Distance covered by each member = 5 km
- Total Distance = Number of members $\times$ Distance per member
- Total Distance = $4 \times 5 \text{ km}$
- Total Distance = $20 \text{ km}$
So, the total distance covered in the race is 20 km.
Step 2: Determine the Total Time
The question states that the total time taken for the race is 30 minutes.
- Total Time = 30 minutes
Step 3: Convert Units for Calculation
The options for average speed are given in units of kilometers per hour ($km/hr$) and meters per second ($m/sec$). Our total distance is in kilometers ($km$), and total time is in minutes. We need to convert the units to match the options.
Converting Time to Hours
To get the speed in $km/hr$, we need to convert the total time from minutes to hours. There are 60 minutes in 1 hour.
$$ \text{Total Time in hours} = \frac{\text{Total Time in minutes}}{\text{Minutes per hour}} $$
$$ \text{Total Time in hours} = \frac{30 \text{ minutes}}{60 \text{ minutes/hour}} $$
$$ \text{Total Time in hours} = 0.5 \text{ hours} $$
Step 4: Calculate Average Speed in km/hr
Now we can calculate the average speed using the total distance in km and total time in hours.
$$ \text{Average Speed} = \frac{\text{Total Distance in km}}{\text{Total Time in hours}} $$
$$ \text{Average Speed} = \frac{20 \text{ km}}{0.5 \text{ hours}} $$
$$ \text{Average Speed} = \frac{20}{0.5} \frac{km}{hr} $$
$$ \text{Average Speed} = 40 \frac{km}{hr} $$
The calculated average speed is 40 km/hr. Let's check if this matches any of the options.
Option 1: 50 km/hr (Does not match)
Option 2: 40 km/hr (Matches!)
Although we have found a matching option, let's quickly calculate the speed in m/sec to see if it matches options 3 or 4.
Converting Distance to Meters and Time to Seconds
- Total Distance = 20 km
- Total Distance in meters = $20 \text{ km} \times 1000 \text{ m/km} = 20,000 \text{ m}$
- Total Time = 30 minutes
- Total Time in seconds = $30 \text{ minutes} \times 60 \text{ seconds/minute} = 1800 \text{ seconds}$
Step 5: Calculate Average Speed in m/sec
$$ \text{Average Speed} = \frac{\text{Total Distance in meters}}{\text{Total Time in seconds}} $$
$$ \text{Average Speed} = \frac{20000 \text{ m}}{1800 \text{ seconds}} $$
$$ \text{Average Speed} = \frac{200}{18} \frac{m}{sec} $$
$$ \text{Average Speed} = \frac{100}{9} \frac{m}{sec} $$
$$ \text{Average Speed} \approx 11.11 \frac{m}{sec} $$
Option 3: 50 m/sec (Does not match)
Option 4: 40 m/sec (Does not match)
Based on our calculations, the average speed is 40 km/hr.
Summary of Calculation Steps
| Step | Description | Calculation | Result |
|---|---|---|---|
| 1 | Calculate Total Distance | $4 \text{ members} \times 5 \text{ km/member}$ | 20 km |
| 2 | Note Total Time | - | 30 minutes |
| 3 | Convert Total Time to Hours | $\frac{30 \text{ minutes}}{60 \text{ minutes/hour}}$ | 0.5 hours |
| 4 | Calculate Average Speed in km/hr | $\frac{20 \text{ km}}{0.5 \text{ hours}}$ | 40 km/hr |
| Optional | Convert Total Distance to Meters | $20 \text{ km} \times 1000 \text{ m/km}$ | 20,000 m |
| Optional | Convert Total Time to Seconds | $30 \text{ minutes} \times 60 \text{ seconds/minute}$ | 1800 seconds |
| Optional | Calculate Average Speed in m/sec | $\frac{20000 \text{ m}}{1800 \text{ seconds}}$ | $\frac{100}{9} \text{ m/sec} \approx 11.11 \text{ m/sec}$ |
Revision Table: Key Concepts
| Concept | Definition/Formula | Units |
|---|---|---|
| Distance | Total length of the path covered. | meters (m), kilometers (km) |
| Time | Duration taken for an event. | seconds (s), minutes (min), hours (hr) |
| Speed | Rate at which an object covers distance. | m/s, km/hr |
| Average Speed | Total distance divided by total time taken. | Same as speed units (m/s, km/hr) |
Additional Information: Understanding Speed and Units
Speed is a measure of how quickly something is moving. It tells us the distance covered per unit of time. The units for speed are derived from the units of distance and time. Common units include meters per second (m/s), kilometers per hour (km/hr), miles per hour (mph), etc.
When solving physics problems involving speed, distance, and time, it is crucial to ensure that the units are consistent. If distance is in kilometers and time is in hours, the speed will be in km/hr. If distance is in meters and time is in seconds, the speed will be in m/s.
Unit conversion is a key skill. Remember the basic conversion factors:
- 1 km = 1000 m
- 1 hour = 60 minutes
- 1 minute = 60 seconds
- 1 hour = 60 minutes $\times$ 60 seconds/minute = 3600 seconds
To convert km/hr to m/s, you can use the conversion factor: $1 \text{ km/hr} = \frac{1000 \text{ m}}{3600 \text{ s}} = \frac{5}{18} \text{ m/s}$.
Conversely, to convert m/s to km/hr, you can use the conversion factor: $1 \text{ m/s} = \frac{18}{5} \text{ km/hr}$.
In this problem, we converted the time to hours to match the km unit of distance, resulting in speed in km/hr, which was one of the required units in the options.
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