An airplane flies from Boston to San Francisco (a distance of 5000 km) in the morning, then immediately returns to Boston. The speed of the plane relative to the air is 240 m/s. The wind is blowing at 41 m/s from west to east, so it is "in the face" of the plane on the way to San Francisco and it is a tail wind on the way back.
(a) What is the average speed of the plane relative to the ground on the way to San Francisco?
(b) What is the average speed relative to the ground on the way back to Boston?
(c) What is the average speed for the entire trip?
From Boston to San Francisco, the speed of the plane relative to the ground is (240 - 41) = 199 m/s
From San Francisco back to Boston, the speed is (240 + 41) = 281 m/s
c) The average speed is just
(199 + 281)/2
To solve this problem, we can use the concept of relative speeds. Let's break down the problem step by step.
(a) To find the average speed of the plane relative to the ground on the way to San Francisco, we need to subtract the wind's speed from the speed of the plane relative to the air.
Relative speed = Speed of plane - Speed of wind
= 240 m/s - (-41 m/s) [Since wind is "in the face" of the plane]
= 240 m/s + 41 m/s
= 281 m/s
Therefore, the average speed of the plane relative to the ground on the way to San Francisco is 281 m/s.
(b) On the way back to Boston, the wind is a tailwind, and hence, it helps the plane move faster. So, we need to add the wind's speed to the speed of the plane relative to the air.
Relative speed = Speed of plane + Speed of wind
= 240 m/s + 41 m/s
= 281 m/s
Therefore, the average speed of the plane relative to the ground on the way back to Boston is 281 m/s.
(c) To find the average speed for the entire trip, we need to take into account the distances covered in each direction. Since the distance from Boston to San Francisco is the same as the distance from San Francisco to Boston (round-trip), the average speed for the entire trip can be calculated by taking the harmonic mean of the speeds on the way to San Francisco and on the way back to Boston.
Average speed for the entire trip = 2 × (Speed to San Francisco × Speed to Boston) / (Speed to San Francisco + Speed to Boston)
= 2 × (281 m/s × 281 m/s) / (281 m/s + 281 m/s)
= 2 × (78961 m^2/s^2) / (562 m/s)
= 280 m/s (rounded to the nearest whole number)
Therefore, the average speed for the entire trip is 280 m/s.
To solve this problem, we need to consider the relative motion of the airplane and the wind. The speed of the airplane relative to the air is given as 240 m/s. However, due to the presence of wind, the airplane's speed relative to the ground will vary.
(a) To find the average speed of the plane relative to the ground on the way to San Francisco, we need to subtract the speed of the wind from the speed of the plane.
Relative speed of the plane to the ground on the way to San Francisco = Speed of the plane relative to the air - Speed of the wind
Relative speed = 240 m/s - (-41 m/s) [Since the wind is blowing from the west to east, it is "in the face" of the plane]
Relative speed = 240 m/s + 41 m/s
Relative speed = 281 m/s
Therefore, the average speed of the plane relative to the ground on the way to San Francisco is 281 m/s.
(b) On the way back to Boston, the wind will now act as a tailwind, which means it will push the plane forward and increase its speed relative to the ground. In this case, we need to add the speed of the wind to the speed of the plane.
Relative speed of the plane to the ground on the way back to Boston = Speed of the plane relative to the air + Speed of the wind
Relative speed = 240 m/s + 41 m/s
Relative speed = 281 m/s
Therefore, the average speed of the plane relative to the ground on the way back to Boston is also 281 m/s.
(c) The average speed for the entire trip can be found by considering the total distance traveled and the total time taken. Since the distance between Boston and San Francisco is 5000 km, and the plane travels back and forth, the total distance covered is 2 * 5000 km = 10000 km.
The total time taken can be calculated by dividing the total distance by the average speed (relative to the ground) of the plane. Since the average speed is 281 m/s, we need to convert the distance to meters.
Total time = (Total distance * 1000) / Average speed
Total time = (10000 km * 1000 m/km) / 281 m/s
Total time = 35587.18 seconds (approximately)
Finally, we can calculate the average speed for the entire trip by dividing the total distance by the total time.
Average speed for the entire trip = Total distance / Total time
Average speed = 10000 km / 35587.18 s
Average speed = 0.28 km/s
Therefore, the average speed for the entire trip is approximately 0.28 km/s.