a plane travels 1.3 km at an angle of 28 degrees to the ground, then changes direction and travels 6.8 km at an angle of 20 degrees to the ground.
A) What is the magnitude of the plane's total displacement? answer in units of km.
B) At what angle above the horizontal is the plane's total displacement? Answer in units of degree.
without time at each angle, one might as well whistle into a hurricane.
Get the times, so you can get the distances. Then break up each distance into horizontal and vertical components, then add horizontal to horizontal, vertical to vertical, and you have the final position.
To solve this problem, we can break down the plane's total displacement into horizontal and vertical components.
Step 1: Find the horizontal and vertical components for each leg of the plane's travel.
Let's start with the first leg of the plane's travel:
Distance traveled: 1.3 km
Angle to the ground: 28 degrees
To find the horizontal component, we can use the equation: horizontal component = distance traveled * cosine(angle)
So, the horizontal component for the first leg is: 1.3 km * cos(28 degrees) = 1.3 km * 0.8829 = 1.148 km
To find the vertical component, we can use the equation: vertical component = distance traveled * sine(angle)
So, the vertical component for the first leg is: 1.3 km * sin(28 degrees) = 1.3 km * 0.4685 = 0.609 km
Similarly, we can find the horizontal and vertical components for the second leg of the plane's travel:
Distance traveled: 6.8 km
Angle to the ground: 20 degrees
The horizontal component for the second leg is: 6.8 km * cos(20 degrees) = 6.8 km * 0.9397 = 6.39 km
The vertical component for the second leg is: 6.8 km * sin(20 degrees) = 6.8 km * 0.3420 = 2.32 km
Step 2: Add up the horizontal and vertical components.
To find the plane's total horizontal displacement, add up the horizontal components from both legs:
Total horizontal displacement = 1.148 km + 6.39 km = 7.538 km
To find the plane's total vertical displacement, add up the vertical components from both legs:
Total vertical displacement = 0.609 km + 2.32 km = 2.929 km
Step 3: Use the Pythagorean theorem to find the magnitude of the plane's total displacement.
The magnitude of the plane's total displacement is given by the square root of the sum of the squares of the horizontal and vertical displacements:
Magnitude of total displacement = √(Total horizontal displacement^2 + Total vertical displacement^2)
Magnitude of total displacement = √(7.538 km^2 + 2.929 km^2)
Magnitude of total displacement = √(56.73 km^2 + 8.577 km^2)
Magnitude of total displacement = √(65.307 km^2)
Magnitude of total displacement ≈ 8.09 km
So, the magnitude of the plane's total displacement is approximately 8.09 km.
Step 4: Calculate the angle above the horizontal for the plane's total displacement.
To find the angle above the horizontal, we can use the inverse tangent function:
Angle above the horizontal = arctan(Total vertical displacement / Total horizontal displacement)
Angle above the horizontal = arctan(2.929 km / 7.538 km)
Angle above the horizontal ≈ 21.91 degrees
Therefore, the plane's total displacement is approximately 8.09 km, and it is at an angle of approximately 21.91 degrees above the horizontal.