An airplane maintains a speed of 597 km/h relative to the air it is flying through as it makes a trip to a city 759 km away to the north.
(a) What time interval is required for the trip if the plane flies through a headwind blowing at 36.0 km/h toward the south?
1 h
(b) What time interval is required if there is a tailwind with the same speed?
2 h
(c) What time interval is required if there is a crosswind blowing at 36.0 km/h to the east relative to the ground?
3 h
To solve this problem, we need to understand the concept of velocity and the effect of winds on an airplane's motion. Let's define some variables to make it easier to solve the problem:
Vp = Velocity of the plane relative to the ground (which depends on the wind)
Va = Velocity of the air the plane is flying through
Vw = Velocity of the wind
d = Distance traveled by the plane
t = Time taken to travel the distance
The velocity of the plane relative to the ground (Vp) can be calculated using the following formula:
Vp = Va + Vw
Now let's solve each part of the problem:
(a) If there is a headwind blowing at 36.0 km/h toward the south, we can substitute the values into the formula:
Vp = 597 km/h (speed of the plane relative to the air) - 36.0 km/h (speed of the headwind)
Vp = 561 km/h
To find the time interval, we can rearrange the formula for velocity:
t = d / Vp
Substituting the values:
t = 759 km / 561 km/h ≈ 1.35 hours
So, the time interval required for the trip is approximately 1.35 hours, which is equal to 1 hour in this case.
(b) If there is a tailwind with the same speed, we can substitute the values into the formula:
Vp = 597 km/h (speed of the plane relative to the air) + 36.0 km/h (speed of the tailwind)
Vp = 633 km/h
Using the rearranged formula for time:
t = d / Vp
Substituting the values:
t = 759 km / 633 km/h ≈ 1.2 hours
So, the time interval required for the trip is approximately 1.2 hours, which is equal to 2 hours in this case.
(c) If there is a crosswind blowing at 36.0 km/h to the east relative to the ground, it does not affect the speed directly. So, the velocity of the plane relative to the ground is still 597 km/h. In this case, the crosswind only affects the direction of the plane, not its speed.
Using the formula again:
t = d / Vp
Substituting the values:
t = 759 km / 597 km/h ≈ 1.27 hours
So, the time interval required for the trip is approximately 1.27 hours, which is equal to 3 hours in this case.
In summary:
(a) With a headwind, the time interval is approximately 1 hour.
(b) With a tailwind, the time interval is approximately 2 hours.
(c) With a crosswind, the time interval is approximately 3 hours.