Backwards Design for Roller Coaster: Projectile Motion

[RockThis52]

For my grade 12 physics class, we need to build a roller coaster. We are given a marble weighing 5 grams and it must be projected at the end of the track. It needs to land 0.5 m at a target when it is projected. Since the coaster is made entirely by myself the ramp in which it will be projected off must be angled and at the exact height in order for the marble to land 0.5 m away. My question is how would I find out the angle and the height of the ramp to project it given only the weight of the marble and the displacement in the x direction. Would it be wiser to figure out the velocity the marble will achieve at that point in the coaster and then adjusting the ramp to land that far away? I have no idea where to start, thanks for any help in advance.

 

[azizlwl]

Think about conservation of energy.
Think about projectile equations.

 

[RockThis52]

Well I know conservation of energy, should I use that before I get to projectile motion? In other words, should I find out the energy and velocity and what not before the projectile than get to the projectile?

 

[rcgldr]

Complicating matters is that the marble will also accumulate angular energy as it rolls, and if the "track" is a pair of rails, the spacing between the rails will effect the ratio of angular to linear velocity.

 

[asteriatic]

I would approach this problem using the concept of backwards design, also known as backward mapping or reverse engineering. This approach involves starting with the desired outcome and working backwards to determine the necessary steps to achieve it.

In this case, the desired outcome is for the marble to land 0.5 m away at the end of the track. To achieve this, we need to determine the necessary angle and height of the ramp to project the marble at the correct velocity and angle.

First, we can use the known displacement in the x direction (0.5 m) to calculate the horizontal component of the marble's velocity using the equation v = d/t, where v is velocity, d is displacement, and t is time. We can assume that the time it takes for the marble to travel the 0.5 m distance is the same as the time it takes for the marble to reach the end of the track, since there will be negligible air resistance. This will give us the minimum velocity required for the marble to land at the target.

Next, we can use the concept of projectile motion to calculate the necessary angle and height of the ramp. We know that the vertical component of the marble's velocity at the end of the track must be zero, since it will be landing horizontally. Using the equation v^2 = u^2 + 2as, where v is final velocity, u is initial velocity, a is acceleration, and s is displacement, we can solve for the initial vertical velocity (u) at the top of the ramp.

Once we have the initial vertical velocity, we can use the equation v = u + at, where v is final velocity, u is initial velocity, a is acceleration, and t is time, to calculate the time it takes for the marble to reach the top of the ramp. This time will also be equal to the time it takes for the marble to reach the end of the track.

Using this time, we can then use the equation s = ut + 0.5at^2, where s is displacement, u is initial velocity, a is acceleration, and t is time, to solve for the necessary height of the ramp.

In summary, to determine the angle and height of the ramp, we need to:

1. Calculate the minimum velocity required for the marble to land at the target using the known displacement in the x direction.

2. Use the concept of projectile motion to calculate the initial vertical velocity (