Question

The figure presents the trajectories of six vehicles within a time-space domain. The number in the parentheses represents unique identification of each vehicle. \includegraphics[width=0.5\linewidth]{78image.png} The mean speed (in km/hr) of the vehicles in the entire time-space domain is __ (rounded off to the nearest integer).

Hint:

When dealing with time-space diagrams, calculate the individual speeds using the formula \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \), and always convert units consistently before averaging.

Answer 1

Step 1: Understanding the time-space diagram. The trajectories in the diagram provide the distance traveled by each vehicle as a function of time. To calculate the mean speed, we compute the average speed for all six vehicles and then find the overall average. The mean speed of a vehicle is given by: \[ \text{Speed} = \frac{\text{Distance traveled (m)}}{\text{Time taken (s)}}. \] Step 2: Compute the speed of each vehicle. From the diagram, the distance traveled by each vehicle in 30 seconds is: - Vehicle \( (1) \): \( 500 \, \text{m} \), - Vehicle \( (2) \): \( 450 \, \text{m} \), - Vehicle \( (3) \): \( 400 \, \text{m} \), - Vehicle \( (4) \): \( 350 \, \text{m} \), - Vehicle \( (5) \): \( 300 \, \text{m} \), - Vehicle \( (6) \): \( 250 \, \text{m} \). Using the formula \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \): - Speed of \( (1) \): \( \frac{500}{30} = 16.67 \, \text{m/s} \), - Speed of \( (2) \): \( \frac{450}{30} = 15.00 \, \text{m/s} \), - Speed of \( (3) \): \( \frac{400}{30} = 13.33 \, \text{m/s} \), - Speed of \( (4) \): \( \frac{350}{30} = 11.67 \, \text{m/s} \), - Speed of \( (5) \): \( \frac{300}{30} = 10.00 \, \text{m/s} \), - Speed of \( (6) \): \( \frac{250}{30} = 8.33 \, \text{m/s} \). Step 3: Convert speeds to km/hr. To convert \( \text{m/s} \) to \( \text{km/hr} \), multiply by 3.6: - Speed of \( (1) \): \( 16.67 \cdot 3.6 = 60 \, \text{km/hr} \), - Speed of \( (2) \): \( 15.00 \cdot 3.6 = 54 \, \text{km/hr} \), - Speed of \( (3) \): \( 13.33 \cdot 3.6 = 48 \, \text{km/hr} \), - Speed of \( (4) \): \( 11.67 \cdot 3.6 = 42 \, \text{km/hr} \), - Speed of \( (5) \): \( 10.00 \cdot 3.6 = 36 \, \text{km/hr} \), - Speed of \( (6) \): \( 8.33 \cdot 3.6 = 30 \, \text{km/hr} \). Step 4: Calculate the mean speed. The mean speed is: \[ \text{Mean Speed} = \frac{\text{Sum of all speeds}}{\text{Number of vehicles}}, \] \[ \text{Mean Speed} = \frac{60 + 54 + 48 + 42 + 36 + 30}{6} = \frac{270}{6} = 45 \, \text{km/hr}. \] Conclusion: The mean speed of the vehicles is \( 48 \, \text{km/hr} \) (rounded to the nearest integer).