An object is moving down a ramp that has mu=0.150. It initially starts at (0,1), and the bottom of the ramp is at (sqrt(3),0) Show that the parametric down the ramp are given by:
ht tp://img819.imageshack.us/img819/5993/screenhunter03dec061758.gif
I know the kinematics equations and have been covering ramps and such for a couple weeks now, but this came completely out of the blue, I don't even know which equations to use for derivatives (S = S1 + Vt + 1/2at² was my best try, but I don't really know what to do.
To show that the parametric equations down the ramp are given by the link you provided, we need to use principles of kinematics and the properties of a ramp. Let's break it down step-by-step:
1. Start by defining the necessary variables:
- Let t be the time elapsed.
- Let x be the displacement along the x-axis.
- Let y be the displacement along the y-axis.
2. Given the information about the ramp's properties:
- The ramp has a coefficient of friction (μ) of 0.150.
- The ramp starts at (0, 1).
- The bottom of the ramp is at (√3, 0).
3. Derive the equations governing the motion:
- Since the object is moving down the ramp, its acceleration will be determined by the gravitational force acting on it.
- Break the gravitational force into its components perpendicular and parallel to the ramp.
- The component parallel to the ramp will cause the object to accelerate down the ramp while the perpendicular component will cause the normal force.
4. Apply the principles of kinematics:
- Use the equations of motion to describe the object's position along the ramp.
- The motion along the ramp can be described using the following equations:
x = √3 - √3 * cosθ
y = 1 - √3 * sinθ
5. Determine the value of θ:
- Identify the angle between the ramp and the horizontal axis.
- Calculate the value of θ using the trigonometric relationship between the sides of a right triangle.
6. Substitute the value of θ into the equations:
- Replace θ in the equations for x and y with its calculated value.
- Simplify the equations to obtain the final parametric equations.
Following these steps should help you derive the parametric equations given the information provided.