Ball rolls of an edge and lands on and inclined plane

In summary, the ball rolls at a constant speed, fast enough to travel straight off the ledge and eventually lands on the inclined plane. The task is to derive an equation for d as a function of v (the initial horizontal speed of the ball) and theta. However, the attempt at a solution is complicated by the fact that the problem is time-independent from the get go.
  • #1
sinisterguy
2
0

Homework Statement


http://img229.imageshack.us/img229/7987/scanik8.jpg [Broken]
the ball rolls at a constant speed, fast enough to travel straight off the ledge and eventually lands on the inclined plane. The task is to derive an equation for d as a function of v (the initial horizontal speed of the ball) and theta.

2. The attempt at a solution
I started by trying to find y by using [tex]\Delta d_{y}= 1/2g \Delta t^2[/tex]
then I moved on to x which is simply [tex]v \Delta t[/tex]
I also know that [tex]\Theta = tan^{-1} (\frac{1/2 g \Delta t^2}{v \Delta t})[/tex]
this is all great, but i wasn't quite sure how to get rid of the t

after some more fiddling i also found that [tex]\Delta t = \frac{d cos \Theta}{v}[/tex], but along that same train of thought, if [tex]d = \sqrt{x^2 + y^2}[/tex] and [tex]x = d cos \Theta[/tex] then that wouldn't work.

my teacher told me i wasn't on the right track so i started over, but i haven't gotten anywhere with that. some help to point me in the right direction would be great
 
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  • #2
I would tackle it using conservation of energy. Such an analysis would be completely time-independent from the get go.
 
  • #3
we haven't learned about conservation of energy yet (well, not enough to be able to apply any mathematical solution to a problem). so that might be a challenge
 
  • #4
OK, then do it using only time-independent kinematic equations. Have you studied projectile motion? If so, then you should have seen an equation for the trajectory of a projectile that doesn't contain t. It is the equation of a parabola that opens downward. You also can write down the equation of the line that contains the inclined plane (remember that the slope of a line is equal to the tangent of its angle of inclination). So basically you can reduce this whole problem to the geometrical problem of finding where the parabola intersects the line.
 

1. How does the angle of the inclined plane affect the distance the ball travels?

The angle of the inclined plane can affect the distance the ball travels in two ways. First, a steeper angle will result in the ball traveling a shorter distance because gravity will act more strongly on the ball, causing it to roll down the incline faster. Second, the angle can also affect the trajectory of the ball, causing it to travel either further or shorter depending on the angle and initial velocity of the ball.

2. What role does friction play in the rolling of the ball?

Friction plays a significant role in the rolling of the ball on an inclined plane. As the ball rolls down the incline, friction between the ball and the surface of the incline will slow the ball down. This friction force is dependent on the materials of the ball and the inclined plane, as well as the magnitude and direction of the ball's motion.

3. How does the mass of the ball affect its motion on the inclined plane?

The mass of the ball affects its motion on the inclined plane in two ways. First, a heavier ball will have a greater gravitational force acting on it, causing it to roll down the incline faster. Second, the mass of the ball can also affect the amount of friction between the ball and the surface of the incline, which can impact the distance and speed of the ball's motion.

4. How does the shape of the ball affect its motion on the inclined plane?

The shape of the ball can affect its motion on the inclined plane in several ways. A spherical ball will have a more predictable and consistent motion compared to a non-spherical ball, which may have different points of contact with the incline. Additionally, the shape of the ball can also impact its rolling resistance and friction with the surface of the incline, affecting its overall motion.

5. Can the initial velocity of the ball affect its motion on the inclined plane?

Yes, the initial velocity of the ball can affect its motion on the inclined plane. The greater the initial velocity, the further the ball will travel before coming to a stop. Also, the direction of the initial velocity can impact the trajectory and distance of the ball's motion on the inclined plane.

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