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
- A car is moving on a horizontal surface in a straight line with a constant velocity of 3 m/s. A ball is thrown vertically upwards at time $t = 0$ from the top of the moving car with a velocity of 20 m/s. The acceleration due to gravity is 10 m/s$^2$.
At what value(s) of time $t$ in second(s), the ball is at a height of 15 m from the top of the moving car? - Velocity of an object fired directly in upward direction is given by $V = 80 – 32 \ t$, where $t$ (time) is in seconds. When will the velocity be between 32 m/sec and 64 m/sec?
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?Two cars start at the same time from the same location and go in the same direction. The speed of the first car is 50 km/h and the speed of the second car is 60 km/h. The number of hours it takes for the distance between the two cars to be 20 km is ___________.
- Trucks (10 m long) and cars (5 m long) go on a single lane bridge. There must be a gap of at least 20 m after each truck and a gap of at least 15 m after each car. Trucks and cars travel at a speed of 36 km/h. If cars and trucks go alternately, what is the maximum number of vehicles that can use the bridge in one hour?
