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 70.0 km/h is able to stay below the airplane. what is the airplane's velocity?
To determine the airplane's velocity, we need to break it down into its horizontal and vertical components.
First, let's consider the woman on the ground and assume that the horizontal distance covered by the airplane is the same as covered by the car. The horizontal component of the airplane's velocity would then match the velocity of the car since they would both be moving at the same speed - 70.0 km/h.
Next, we need to determine the vertical component of the airplane's velocity. The angle of ascent, 35 degrees above the horizontal, gives us the information required.
Since the angle of ascent is given, we can use trigonometric relationships to find the vertical component of the airplane's velocity. The vertical component can be determined using the sine function:
sin(theta) = opposite/hypotenuse
Here, the vertical component of the airplane's velocity is the opposite side, and the hypotenuse is the total velocity of the airplane.
sin(35 degrees) = vertical component/hypotenuse
To solve for the vertical component of the velocity, multiply both sides by the hypotenuse:
hypotenuse * sin(35 degrees) = vertical component
Now, we need to find the hypotenuse (total velocity) of the airplane. To do this, we can use the Pythagorean theorem:
hypotenuse^2 = horizontal component^2 + vertical component^2
Substituting the values we have:
hypotenuse^2 = (70.0 km/h)^2 + (hypotenuse * sin(35 degrees))^2
Now, solve the equation to find the hypotenuse (total velocity) of the airplane:
(70.0 km/h)^2 = hypotenuse^2 - (hypotenuse * sin(35 degrees))^2
Solve this equation to find the value of hypotenuse (the total velocity) of the airplane. In this case, the hypotenuse will represent the velocity of the airplane.