A jet plane is flying horizontally with a speed of 500 m/s over a hill that slopes upward with a 3% grade (i.e., the "rise" is 3% of the "run"). What is the component of the plane's velocity perpendicular to the ground? (Assume the +x-direction is up along the slope and the +y-direction is perpendicular to the slope and out of the ground.)
Let v=ground velocity of plane = 500 m/s
Since slope is rise/run, so the angle of the "ground" relative to the horizontal
θ=tan<-1>(3/100)
Component of velocity vector perpendicular to the ground
= v*sin(θ)
= ?
That didn't work, I already tried that. I only have 1 more try left so I need to get the right answer this time haha
tan theta = 3/100 so sin theta = .0299865
-500 (.0299865) = -14.99
they defined y as up so it is negative
Thank you, that worked!!
You are welcome.
Good catch, didn't see the details!
Why is it negative?
"Assume the +x-direction is up along the slope and the +y-direction is perpendicular to the slope and out of the ground."
Since the perpendicular component of the velocity "digs" into the ground, so by definition of the y-axis, it is negative.
To find the component of the plane's velocity perpendicular to the ground, we can break down the velocity vector into its parallel and perpendicular components.
Step 1: Calculate the incline angle:
The slope of the hill is given as a percentage, which represents the rise over run. In this case, the rise is 3% of the run. To find the incline angle, we can use the inverse tangent function (arctan) with the rise/run ratio.
tan(incline angle) = rise / run
tan(incline angle) = 3 / 100
Now, we can find the incline angle (θ) by taking the arctan of both sides:
incline angle = arctan(3 / 100)
Step 2: Calculate the parallel component of velocity:
The velocity of the plane is given as 500 m/s horizontally. The parallel component of velocity (V_parallel) is the component that aligns with the incline, which in this case is along the slope. Since the slope is upward, the positive x-direction is along the slope.
V_parallel = velocity * cos(incline angle)
V_parallel = 500 m/s * cos(incline angle)
Step 3: Calculate the perpendicular component of velocity:
The perpendicular component of velocity (V_perpendicular) is the component that is perpendicular to the incline and goes into the ground. In this case, the positive y-direction is perpendicular to the slope and out of the ground.
V_perpendicular = velocity * sin(incline angle)
V_perpendicular = 500 m/s * sin(incline angle)
By substituting the value of the incline angle from Step 1 into these equations, we can calculate the parallel and perpendicular components of velocity.