You drive on an interstae 10 from San Antonio to Huston, half the [b]time[/b] at 55km/h and the other half at 90km/h. On the way back you travel half the [b]distance[/b] at 55km/h and the other half at 90km/h.
What is your average speed from San Antonio to Houston
What is your speed from houston to San Antonio?
c) speed for entire trip
d) What is your average velocity for the whole trip?
Let T = total time of trip and D = toal distance covered
average speed = distance/time
(a) SA to Houston:
V1 = 55*(T/2)+90*(T/2) / T = (55+90)/2) = ?
(b) Houston to SA:
V2 = D/[(D/2)(1/55)+(D/2)(1/90)]
= 2/[(1/55) + (1/90)]
(c) 2D/[D/V1 + D/V2]=
2/[1/V1 = + 1/V2)]
(d) Think about the definition of velocity. You end up where you started in this case. Vector displacement = 0.
72.5 km/hr
To calculate the average speed from San Antonio to Houston, you will need to find the total time taken for the trip and the total distance covered.
Let T be the total time of the trip and D be the total distance covered.
(a) Average speed from San Antonio to Houston:
In this case, you drive half the time at 55 km/h and the other half at 90 km/h. The total time taken is T, so the time driving at 55 km/h is T/2 and the time driving at 90 km/h is also T/2.
The distance covered at 55 km/h is (55 km/h) * (T/2) and the distance covered at 90 km/h is (90 km/h) * (T/2).
So, the total distance covered is (55 km/h) * (T/2) + (90 km/h) * (T/2)
To calculate the average speed, you divide the total distance by the total time:
Average speed = (55 km/h) * (T/2) + (90 km/h) * (T/2) / T
Simplifying the expression, you get:
Average speed = (55 km/h + 90 km/h) / 2
(b) To calculate the speed from Houston to San Antonio, you can use the same approach. Since the distance covered is the same as the distance from San Antonio to Houston, which we will call D, the times taken will be different due to the different speeds.
The time taken at 55 km/h is D / (2 * 55 km/h) and the time taken at 90 km/h is D / (2 * 90 km/h).
Speed from Houston to San Antonio = D / [(D / 2) * (1 / 55 km/h) + (D / 2) * (1 / 90 km/h)]
Simplifying the expression, you get:
Speed from Houston to San Antonio = 2 / [(1 / 55 km/h) + (1 / 90 km/h)]
(c) For the speed for the entire trip, you can use the formula:
Speed for entire trip = 2D / [D / V1 + D / V2]
Where V1 is the average speed from San Antonio to Houston and V2 is the speed from Houston to San Antonio.
Simplifying the expression, you get:
Speed for entire trip = 2 / [1 / V1 + 1 / V2]
(d) Finally, the average velocity for the whole trip is determined by the definition of velocity. Since you end up where you started, the vector displacement is 0. Therefore, the average velocity for the whole trip is also 0.