A woman on the ground sees an airplane climbing at an angle of 35o above the horizontal. She gets into her car and by driving at 70.0 km/h is able to stay directly below the airplane. What is the airplane’s velocity?
76.05km/h
To find the airplane's velocity, we can break it down into its horizontal and vertical components.
1. Horizontal component:
The horizontal component of velocity remains constant since the airplane is moving horizontally above the ground. Therefore, the horizontal component of the airplane's velocity will be the same as the car's velocity.
Given:
Car's velocity (horizontal component) = 70.0 km/h
2. Vertical component:
The vertical component of velocity is related to the angle at which the airplane is climbing.
Given:
Angle of climb = 35°
We can use trigonometry to find the vertical component of velocity:
Vertical component = Cos(angle) * Total velocity
Since the vertical component is "climbing," we use the cosine function. The total velocity is the magnitude of the airplane's velocity, which can be found using Pythagorean theorem.
Total velocity = sqrt((Horizontal component)^2 + (Vertical component)^2)
Rearranging the equation, we get:
(Vertical component)^2 = (Total velocity)^2 - (Horizontal component)^2
Now, substitute the given values:
(Vertical component)^2 = (Total velocity)^2 - (70.0 km/h)^2
Finally, we can find the vertical component of velocity:
Vertical component = sqrt((Total velocity)^2 - (70.0 km/h)^2)
3. Combine horizontal and vertical components:
Once we have both the horizontal and vertical components of velocity, we can find the vector sum of these two components using Pythagorean theorem:
Total velocity = sqrt((Horizontal component)^2 + (Vertical component)^2)
Substitute the known values:
Total velocity = sqrt((70.0 km/h)^2 + (Vertical component)^2)
Now we can substitute the value of the vertical component we found earlier.
Total velocity = sqrt((70.0 km/h)^2 + ((Total velocity)^2 - (70.0 km/h)^2))
Simplify the equation:
Total velocity = sqrt((Total velocity)^2)
Since the magnitude of the velocity is always positive, we can remove the square root:
Total velocity = |Total velocity|
Therefore, the magnitude of the airplane's velocity is equal to the car's velocity, which is 70.0 km/h.
In conclusion, the airplane's velocity is 70.0 km/h.
To find the airplane's velocity, we need to consider its horizontal and vertical components separately. Let's break down the problem step by step:
Step 1: Find the horizontal component of the airplane's velocity.
Since the airplane is climbing at an angle of 35 degrees above the horizontal, we can use trigonometry to find its horizontal component. The horizontal component is the adjacent side of the angle.
Horizontal component = Cos(angle) * Total velocity
The total velocity is not explicitly given in the question, but we can find it from the information given. The woman in the car is driving directly below the airplane. Therefore, her velocity must match the horizontal component of the airplane's velocity.
So, the horizontal component of the airplane's velocity is equal to the woman's velocity, which is 70.0 km/h.
Step 2: Find the vertical component of the airplane's velocity.
The vertical component is the opposite side of the angle.
Vertical component = Sin(angle) * Total velocity
Again, the total velocity is not explicitly given, but we can use the fact that the airplane's velocity will remain constant, and it will maintain a constant climb angle. The vertical component of the airplane's velocity must match the vertical velocity of the woman's car, which is 0 since the car is driving horizontally.
Therefore, the vertical component of the airplane's velocity is 0.
Step 3: Combine the horizontal and vertical components to find the airplane's velocity.
The total velocity of the airplane can be found using the Pythagorean theorem:
Total velocity = √(Horizontal component^2 + Vertical component^2)
Substituting the values we found in step 1 and step 2:
Total velocity = √(70.0 km/h)^2 + (0)^2
= √(70.0 km/h)^2
= 70.0 km/h
So, the airplane's velocity is 70.0 km/h.
v cos35° = 70
now just find v