A car covers the first 41 km of its journey in 45 min and covers the remaining 23 km in 35 min. What is the average speed (in m/sec) of the car?
13 \(\frac{1}{3}\)
Calculating Average Speed in m/sec
The question asks us to find the average speed of a car for its entire journey. The journey is divided into two parts, and we are given the distance and time for each part. Average speed is defined as the total distance covered divided by the total time taken for the journey.
Understanding Average Speed
Average speed is not just the average of the speeds in different segments. It considers the total distance traveled over the total duration of the travel. The formula is:
$$ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} $$
Breaking Down the Journey
The journey has two distinct segments:
- Segment 1: Distance = 41 km, Time = 45 min
- Segment 2: Distance = 23 km, Time = 35 min
Calculating Total Distance and Total Time
First, let's find the total distance covered by the car and the total time taken for the journey.
- Total Distance: Sum of distances of Segment 1 and Segment 2.
- Total Time: Sum of times taken for Segment 1 and Segment 2.
$$ \text{Total Distance} = 41 \text{ km} + 23 \text{ km} = 64 \text{ km} $$
$$ \text{Total Time} = 45 \text{ min} + 35 \text{ min} = 80 \text{ min} $$
Converting Units to m/sec
The question asks for the average speed in meters per second (m/sec). We need to convert the total distance from kilometers to meters and the total time from minutes to seconds.
- To convert kilometers to meters, multiply by 1000 (since 1 km = 1000 m).
- To convert minutes to seconds, multiply by 60 (since 1 min = 60 sec).
Conversion of Total Distance:
$$ \text{Total Distance} = 64 \text{ km} \times 1000 \frac{\text{m}}{\text{km}} = 64000 \text{ m} $$
Conversion of Total Time:
$$ \text{Total Time} = 80 \text{ min} \times 60 \frac{\text{s}}{\text{min}} = 4800 \text{ s} $$
Calculating Average Speed in m/sec
Now we can use the average speed formula with the total distance in meters and the total time in seconds.
$$ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{64000 \text{ m}}{4800 \text{ s}} $$
Let's simplify the fraction:
$$ \text{Average Speed} = \frac{64000}{4800} \frac{\text{m}}{\text{s}} = \frac{640}{48} \frac{\text{m}}{\text{s}} $$
Divide both numerator and denominator by common factors. Both are divisible by 16:
$$ \frac{640 \div 16}{48 \div 16} = \frac{40}{3} \frac{\text{m}}{\text{s}} $$
The average speed is $\frac{40}{3}$ m/sec. We can express this as a mixed number:
$$ \frac{40}{3} = 13 \frac{1}{3} $$
So, the average speed is $13 \frac{1}{3}$ m/sec.
Summary of Calculation Steps
| Step | Description | Value |
|---|---|---|
| 1 | Calculate Total Distance | $41 \text{ km} + 23 \text{ km} = 64 \text{ km}$ |
| 2 | Calculate Total Time | $45 \text{ min} + 35 \text{ min} = 80 \text{ min}$ |
| 3 | Convert Total Distance to Meters | $64 \text{ km} \times 1000 \frac{\text{m}}{\text{km}} = 64000 \text{ m}$ |
| 4 | Convert Total Time to Seconds | $80 \text{ min} \times 60 \frac{\text{s}}{\text{min}} = 4800 \text{ s}$ |
| 5 | Calculate Average Speed (Total Distance / Total Time) | $\frac{64000 \text{ m}}{4800 \text{ s}} = \frac{40}{3} \frac{\text{m}}{\text{s}}$ |
| 6 | Express Average Speed as Mixed Number | $\frac{40}{3} \frac{\text{m}}{\text{s}} = 13 \frac{1}{3} \frac{\text{m}}{\text{s}}$ |
The calculated average speed is $13 \frac{1}{3}$ m/sec.
Revision Table: Average Speed Concepts
| Concept | Definition | Formula |
|---|---|---|
| Speed | Rate at which an object covers distance. | $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$ |
| Average Speed | Total distance covered divided by the total time taken. Useful when speed varies. | $\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$ |
| Units of Speed | Common units include meters per second (m/s), kilometers per hour (km/h), miles per hour (mph). | Conversion factors needed to switch between units (e.g., 1 km = 1000 m, 1 hour = 3600 s). |
Additional Information: Speed vs. Velocity
While often used interchangeably in everyday language, speed and velocity are distinct concepts in physics:
- Speed: A scalar quantity that measures how fast an object is moving. It only has magnitude (e.g., 10 m/s).
- Velocity: A vector quantity that measures how fast an object is moving and in what direction. It has both magnitude and direction (e.g., 10 m/s North).
Average speed considers the total distance traveled, regardless of direction changes. Average velocity considers the total displacement (change in position from start to end) divided by total time, which can be zero if the object returns to its starting point.
This problem specifically asks for average speed, so we used the total distance traveled.
Important Questions from Average Speed
Akhil drives a car from his home to office at an average speed of 50 km/h and reaches office 10 minutes early. But one day due to some problem with the car, he could drive at an average speed of 30 km/h only and reached office 10 minutes late. How far is his office from home ?
During a flight of 900 km, an aircraft was slowed down due to bad weather. Its average speed for the trip was reduced by 300 km/h and the time of flight increased by 30 min. The original duration of the flight was :
If a man travels from A to B at a speed of 50 km/h and returns by increasing his speed by 40%, then find his average speed (to 2 decimal places) for both the trips.
A man travels a distance of 420 km by train which moves at the speed of 75 km/h and returns back by car at the speed of 50 km/h. Find his average speed for the whole journey.
A train runs at a speed of 90 km/h in the first 10 minutes and 60 km/h in next 25 minutes and 15 km/h in last 4 minutes. What is the average speed of the train?
