52. Bryan Allen pedaled a human-powered aircraft across the English Channel from the cliffs of Dover to Cap Gris-Nez on June 12, 1979. (a) He flew for 169 min at an average velocity of 3.53 m/s in a direction 45º south of east. What was his total displacement? (b) Allen encountered a headwind averaging 2.00 m/s almost precisely in the opposite direction of his motion relative to the Earth. What was his average velocity relative to the air? (c) What was his total displacement relative to the air mass?
A)
Known
t = 169min
Va = 3.53m/s
Theta= 45°
x=?
169*60 = 10140
10140*3.53 = x
x= 35794.2m at 45° South of East
B)
3.53m/s is the velocity of Bryan 2 m/s is the velocity of the air. Ignoring air resistance we are comparing the velocity of that mass of air relative to the ground and Bryan relative to the ground to each other so instead of subtracting the two values we add them.
Degree is determined by reference angle
3.53+2
= 5.53m/s at 45° North of west
C)
Now just multiply by time
10140*5.53 = x
56074.2m
To solve these questions, we will break them down step-by-step.
(a) To find Bryan Allen's total displacement, we can use the formula:
Displacement = Velocity * Time
The velocity is given as 3.53 m/s, and the time is given as 169 minutes (but we need to convert it to seconds since velocity is given in meters per second). So:
Time in seconds = 169 minutes * 60 seconds/minute = 10,140 seconds
Displacement = 3.53 m/s * 10,140 s = 35,768.2 meters
However, we need to find the total displacement, considering both magnitude and direction. The direction is given as 45º south of east. To simplify the calculation, we can find the x and y components of the displacement, and then combine them.
x-component = Displacement * cos(45º) = 35,768.2 meters * cos(45º) = 25,327.9 meters
y-component = Displacement * sin(45º) = 35,768.2 meters * sin(45º) = 25,327.9 meters
So, the total displacement is (25,327.9 m, 25,327.9 m) south-east of the starting point.
(b) To find his average velocity relative to the air, we need to take into account the headwind. We can subtract the headwind velocity from his average velocity.
Average velocity relative to the air = Average velocity - Headwind velocity
Average velocity relative to the air = 3.53 m/s - 2.00 m/s = 1.53 m/s
So, Bryan Allen's average velocity relative to the air is 1.53 m/s in the direction 45º south of east.
(c) The total displacement relative to the air mass will be the same as the total displacement, as the wind does not affect the overall displacement. So, the total displacement relative to the air mass is (25,327.9 m, 25,327.9 m) south-east of the starting point.
To calculate Bryan Allen's total displacement, average velocity relative to the air, and total displacement relative to the air mass, we'll need to break down the given information and use trigonometry.
(a) Total Displacement:
To find his total displacement, we need to calculate the horizontal and vertical components separately and then combine them using vector addition.
Given information:
- Average velocity: 3.53 m/s
- Direction: 45º south of east
- Time: 169 minutes (convert to seconds: 169 x 60 = 10,140 s)
First, we'll find the horizontal component of the velocity:
Velocity (horizontal) = average velocity * cos θ
= 3.53 m/s * cos 45º
= 3.53 m/s * (√2/2)
= 2.49 m/s
Next, we'll find the vertical component of the velocity:
Velocity (vertical) = average velocity * sin θ
= 3.53 m/s * sin 45º
= 3.53 m/s * (√2/2)
= 2.49 m/s
Since Bryan Allen flew for 169 minutes, we'll assume he maintained a constant velocity throughout his flight. So, the total displacement is given by:
Total Displacement = velocity * time
= (2.49 m/s, horizontal) * 10,140 s + (2.49 m/s, vertical) * 10,140 s
= (2.49 m/s * 10,140 s, horizontal) + (2.49 m/s * 10,140 s, vertical)
= (25,219.6 m, horizontal) + (25,219.6 m, vertical)
= (25,219.6 m, 25,219.6 m)
= (35,618 m, 45º south of east)
Therefore, Bryan Allen's total displacement is 35,618 meters in a direction 45º south of east.
(b) Average Velocity relative to the air:
To find his average velocity relative to the air, we need to subtract the headwind's velocity from his average velocity.
Given information:
- Headwind velocity: 2.00 m/s in the opposite direction
Average Velocity relative to the air = average velocity - headwind velocity
= 3.53 m/s - 2.00 m/s
= 1.53 m/s
Therefore, his average velocity relative to the air is 1.53 m/s.
(c) Total Displacement relative to the air mass:
Since the headwind affected his motion, the total displacement relative to the air mass will be different from the total displacement calculated in part (a). To find the total displacement relative to the air mass, we'll need to consider the headwind's effect on both the horizontal and vertical components.
Given information:
- Headwind velocity: 2.00 m/s in the opposite direction
Total Displacement relative to the air mass = Total Displacement - Headwind displacement
The horizontal component of the headwind's displacement can be calculated as:
Headwind displacement (horizontal) = Headwind velocity * cos 180º
= 2.00 m/s * (-1)
= -2.00 m/s
The vertical component of the headwind's displacement can be calculated as:
Headwind displacement (vertical) = Headwind velocity * sin 180º
= 2.00 m/s * 0
= 0 m/s
Therefore, the total displacement relative to the air mass is:
Total Displacement relative to the air mass = (Total Displacement (horizontal) - Headwind displacement (horizontal)) + (Total Displacement (vertical) - Headwind displacement (vertical))
= (2.49 m/s * 10,140 s - (-2.00 m/s * 10,140 s), horizontal) + (2.49 m/s * 10,140 s - 0 m/s * 10,140 s, vertical)
= (25,219.6 m + 20,280 m, horizontal) + (25,219.6 m + 0 m/s, vertical)
= (45,499.6 m, horizontal) + (25,219.6 m, vertical)
= (56,747.05 m, 42.5º south of east)
Therefore, his total displacement relative to the air mass is approximately 56,747.05 meters in a direction 42.5º south of east.
A. (169 * 60) seconds * 3.53 meters/second = 357,942 meters distance which is 357,942 meters at 45 degrees S of E. That is 253,064 meters east and 253,064 meters south.
B. 3.53 - 2.00 = 1.53 m/s
C. is the same as A, 3.53 m/s SE. He just does not get to France