What is the force required and amount of work done to stop this motion?

[5hassay]

Homework Statement

An object of mass 1200 kg rests at a height of 95 m above the ground on a roller coaster. After a series of inclined planes, repeately reaching a height of zero, (bumps and drops of the roller coaster) the roller coaster lays horizontal on the ground (height zero) for 7 m until it (the diagram) ends. What would be the applied force required to end the motion of this object at the absolute ending point of the roller coaster if it were to ride the entire coaster, and the related work required to end this motion? No friction exists, and this is an isolated system.

Homework Equations

Total Energy (E*T) = Potential Energy (E*p) + Kinetic Energy (E*k)
Total Energy 1 [E*(T-1)] = Total Energy 2 [E*(T-2)]
Work (W) = Applied Force (F*A) X cos Angle (O) X delta displacement (^d)
E*p = mass (m) X acceleration due to gravity (g) X ^h
E*k = 1/2 X m X speed (v) squared

The Attempt at a Solution

1. Calculate the total energy at the beginning of the system, for it will be equal to the total energy of the system when the ride begins to end and height is zero (before the 7 m length), for that will be the amount of energy required to be stopped to end motion.

E*(T-1) = E*p + E*k
The object is at rest at this point, thus
E*(T-1) = E*p; = 1200 kg X 9.8 m/s^2 X 95 m; = 1,117,200 J.
thus, E*(T-2) = 1,117,200 J
where, E*(T-2) is such total energy of the system immediately as height becomes zero

2. To end the motion of the system at this point (where the roller coaster ends, height is zero, at the beginning of the 7 m length until the absolute end) exactly 1,117,200 J
of work must be done, over a displacement of 7 m. Then, solve for the relevant applied force.

W = F*A X cos O X ^d
the object is displacing in the direction of motion, thus
O = 0 degrees. Thus,
W = F*A X ^d. Thus,
F*A = W / ^d; = 1,117,200 J / 7 m; = 159,600 N.

>>>Therefore, to end this motion of the object, 1,117,200 J of work is required (equal to the total energy of the system) and 159,600 N of applied force (over a 7 m displacement).


PROBLEM: The question asks for the applied force required to end motion of the object at a point Z, which is the absolute ending of the roller coaster. I was confused on how to calculate, but thought the work equation was a logical solution. Then, I was confused on what to do with the relevant displacement in the equation and the 7 m length of the roller coaster where height is zero before it ends. Thus, I would like to know if I strayed in any theory, if I am going about it wrong, et cetera.

Much appreciation!

 

[anathema]

Dear forum post,

Your approach to solving this problem is correct. However, there are a few things to consider in order to find the applied force required at the absolute ending point of the roller coaster.

First, it is important to note that the object's total energy is equal to its potential energy at the beginning of the system (when it is at a height of 95 m) and its kinetic energy at the end of the system (when it is at a height of 0 m). This means that the total energy at the end of the system is not equal to the total energy at the beginning, as you have calculated. Instead, it should be:

E*(T-2) = E*p + E*k
= (1200 kg)(9.8 m/s^2)(0 m) + (1/2)(1200 kg)(v^2)
= 600v^2 J

Where v is the speed of the object at the end of the system.

Next, in order to find the applied force required to stop the object at the absolute ending point, we need to consider the work done on the object. This work is equal to the change in the object's kinetic energy, which is equal to the total energy at the end of the system (E*(T-2)).

W = E*(T-2)
= 600v^2 J

Now, we need to determine the relevant displacement in the equation. This is the distance traveled by the object from the end of the 7 m horizontal section to the absolute ending point. Let's call this distance x. Then, the total displacement is 7 m + x. So, the work done on the object is:

W = F*A x (7 m + x)

Finally, we can plug in the values and solve for the applied force:

600v^2 J = F*A x (7 m + x)
F*A = 600v^2 J / (7 m + x)

Therefore, the applied force required to stop the object at the absolute ending point of the roller coaster is:

F*A = 600v^2 J / (7 m + x)

I hope this helps clarify the solution for you. Let me know if you have any further questions.