To find the average speed for the entire journey, we need to consider the total distance traveled and the total time taken. The average speed is not simply the arithmetic mean of the two speeds because the vehicle spends different amounts of time traveling at each speed.
Calculating Average Speed for Automobile Journey
Let the distance between city A and city B be represented by '$d$' km.
- Onward Journey:
- Speed = 60 km/h
- Time taken ($t_1$) = Distance / Speed = $\frac{d}{60}$ hours
- Return Journey:
- Speed = 90 km/h
- Time taken ($t_2$) = Distance / Speed = $\frac{d}{90}$ hours
Total Distance and Time Calculation
- Total Distance traveled = Distance (A to B) + Distance (B to A) = $d + d = 2d$ km.
- Total Time taken = Time ($t_1$) + Time ($t_2$) = $\frac{d}{60} + \frac{d}{90}$ hours.
- To add the fractions for time:
- Find a common denominator for 60 and 90, which is 180.
- $\frac{d}{60} + \frac{d}{90} = \frac{3d}{180} + \frac{2d}{180} = \frac{5d}{180}$ hours.
Average Speed Formula and Result
The formula for average speed is:
$ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} $Substitute the values:
$ \text{Average Speed} = \frac{2d}{\frac{5d}{180}} $Simplify the expression:
$ \text{Average Speed} = 2d \times \frac{180}{5d} $The '$d$' cancels out:
$ \text{Average Speed} = \frac{2 \times 180}{5} = \frac{360}{5} = 72 \text{ km/h} $Therefore, the average speed for the entire journey is 72 km/h.
Important Questions from Speed Distance and Time
- Two cars, P and Q, start from a point X in India at 10 AM. Car P travels North with a speed of 25 km/h and car Q travels East with a speed of 30 km/h. Car P travels continuously but car Q stops for some time after travelling for one hour. If both the cars are at the same distance from X at 11:30 AM, for how long (in minutes) did car Q stop?
In a 500 m race, P and Q have speeds in the ratio of 3: 4. Q starts the race when P has already covered 140 m.
What is the distance between P and Q (in m) when P wins the race?A vehicle is moving at a speed of 12 m/s on a level road. It applies emergency brakes and starts to skid without rolling in a straight path. The deceleration of the vehicle is constant after braking and it comes to rest at a distance of 15 m. Assuming, $g = 10$ m/s$^2$, the coefficient of kinetic friction between the tyres and road is _________ [round off to 2 decimal places]
- The distance between Delhi and Agra is 233 km. A car P started travelling from Delhi to Agra and another car Q started from Agra to Delhi along the same road 1 hour after the car P started. The two cars crossed each other 75 minutes after the car Q started. Both cars were travelling at constant speed. The speed of car P was 10 km/hr more than the speed of car Q. How many kilometers the car Q had travelled when the cars crossed each other?
Two trains started at 7AM from the same point. The first train travelled north at a speed of 80km/h and the second train travelled south at a speed of 100 km/h. The time at which they were 540 km apart is _______________ AM.
