31.3. Space mean speed (v_s)

The space mean speed averages the spot speed, but spatial weightage is given instead of temporal. This is derived as below. Consider unit length of a road, and let v_i be the spot speed of i-th vehicle. Let t_i be the time the vehicle takes to complete unit distance and is given by 1/v_i. If there are n such vehicles, then the average travel time t_s is given by,

Σt_i = 1/n Σ (1/v_i) (31.3)

Space mean speed focuses on how quickly vehicles travel over a distance of road rather than time. To understand this, it's essential to consider a road of one unit length and multiple vehicles each going at different speeds. For each vehicle, the time it takes to travel that unit distance is the reciprocal of its speed (1 divided by the speed). The average travel time across all vehicles is then calculated. This means that instead of averaging speeds directly, we first determine the time each vehicle takes, giving us a different perspective on speed.

Imagine a group of runners, each at a different speed, racing a 100-meter dash. If we want to know how fast the group is moving as a whole, we can't simply average their running speeds. Instead, we have to consider how long each runner takes to finish the race. The slower runners will naturally take up more time on the track, so they have a greater impact on the group's overall speed, much like how slower cars on a busy road affect average traffic flow.

If t is the average travel time, then average speed is given by v_s = 1/t_s. Therefore from the above equation, v_s = n / Σ(1/v_i) (31.4). This is simply the harmonic mean of the spot speed.

The average speed calculated via space mean speed is derived from the inverse of the average travel time. In simpler terms, once we find the total time taken by all vehicles, we take the reciprocal to find speed. The mathematical expression we end up with approximates the harmonic mean of individual speeds. This is important because it emphasizes the role of slower speeds in determining the average speed of all vehicles traveling a stretch of road.

Think about filling different-sized containers with water from a faucet. If one container is much smaller and fills up slowly while others are larger, the overall rate of water flowing through might be more affected by the slower filling container. In traffic terms, the slower vehicles on the road fill the 'space' more significantly, just like that slower container, thus affecting the calculated average speed more than the faster ones.

If the spot speeds are expressed as a frequency table, then, v_s = Σ(q_i * v_i) / Σ(q_i) (31.5), where q_i is the number of vehicles having speed v_i and n is the number of such observations.

When data is collected in the form of a frequency table, where speeds are represented alongside the number of vehicles at that speed, space mean speed can be calculated using a weighted average. Here, the vehicle count at each speed acts like weights. By multiplying the speed by the count of vehicles and summing these products, then dividing by the total number of vehicles, we find the average speed that accounts for how many vehicles are moving at each speed.

Consider a teacher grading a class on different assignments, some with more students than others. If more students performed very well on one assignment than another, those grades would contribute more heavily to the overall class average, similar to how more vehicles at a particular speed affect the space mean speed.

Example 1 If the spot speeds are 50, 40, 60, 54, and 45, then find the time mean speed and space mean speed.

Solution: Time mean speed v_t = Σv_i / n = 249 / 5 = 49.8. Space mean speed is the harmonic mean of spot speed. Therefore, v_s = n / Σ(1/v_i) = 5 / (1/50 + 1/40 + 1/60 + 1/54 + 1/45) = 48.82.

Example 2 The results of a speed study is given in the form of a frequency distribution table. Find the time mean speed and space mean speed.

Examples provide practical applications of the theory. In Example 1, the individual speeds are used to calculate the time mean speed by simply averaging. The same data is then used to find the space mean speed using the harmonic mean, which emphasizes slower speeds. Example 2 would likely involve calculations from a frequency table, showcasing how to adapt the concept into real-world datasets effectively.

Think of a cooking recipe where the cooking times for different ingredients vary. If you average the cooking times, it gives a simple mean, but if you focus on how long each ingredient takes individually, the longest cooking time creates the main influence on the overall time needed, just like how slower vehicle speeds affect the overall average speed on a road.

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