Okay, So we are going over vectors in class, given Cartesian coordinates and convert them to polar, and vice-versa.
So the question is that a skateboard rolls up a ramp shown in the image shown, at a initial speed of $85.0\:cm/s$ and slows down at $4.15\:m/s^2$. What is the initial velocity? Here is the image:
I honestly have no idea what to do, and my book doesn't seem to explain anything. I do know that $velocity\:=\:speed\:*\:direction$ and the book says (How do you find the angle from the diagram?) but I have no idea what to do.
So how do I find the initial velocity, and how do I find the acceleration (it's slowing down)?
This can be in either polar or Cartesian coordinates.
Best Answer
If the ramp were horizontal, the skateboard wouldn't be slowing down at all, right? I.e. the acceleration would be 0 $m/s^2$. If the ramp were vertical, the skateboard would be "slowing down" at a rate of $-9.8 m/s^2$. In our case, the acceleration is $-4.15 m/s^2$, and the idea is that this number must have something to do with the angle of the ramp.
What we need to do is consider the vector that represents the acceleration of the object, due to gravity, and then decompose it into two perpendicular components, one perpendicular to the ramp's surface (which is cancelled by the normal force, this is Newton's third law), and one parallel to the ramp, which is responsible for the actual change in velocity.
If we call the angle of the ramp $\theta$, you can determine that one component is related to $\cos(\theta)$ and the other to $\sin(\theta)$. I'll leave the details to you to complete. You should also be able to verify that this fits with the horizontal/vertical example we started with; i.e. if $\theta = 0$ (horizontal ramp), the component of acceleration parallel to the surface of the ramp should be $0$, and if $\theta = 90\deg$, it has magnitude $9.8$.
Here's a picture illustrating the decomposition: