Question

"the splash mountain flume ride at Walt disney world's magic kingdom in orlando, florida boats use one of the steepest flume ride in the world. The boats plunge 52 feet down a 43-degree incline near the final stage of the ride. Assuming a coefficient of friction pf 0.12, determine the acceleration of the boat and riders."

Answer 1

a= 5.8 m/s²

Step-by-step explanation:

Let's give a look to the external forces acting on the boat:

1. Gravity (Fg) downward.
2. Normal Force (perpendicular to the surface of the incline)
3. Friction Force (a fraction of the normal force, being the proportionality constant the coefficient of friction) opposing to the movement along the incline.

We can apply Newton's 2nd law to two axis mutually perpendicular, let's call X to an axis parallel to the incline, and Y to one perpendicular to it.

The sum of forces along the axis Y, must be 0, as there is no net force in this direction (the boat doesn't move through the incline), so we can write:

Fnet (Y) = Fn- Fg* cos 43° = 0⇒ Fn = Fg* cos 43º =m*g*cos 43º

In the direction parallel to the incline, there are two forces acting: the component of Fg parallel to the incline (in the direction of the movement) and the friction force, opposing to it.

Fnet (X) = m*g*sin 43º - μk* N = m*ax

Now, we can take the value of N that we have already got from the equation for the other axis:

Fnet(X) = m*g*sin 43º - μk*m*g*cos 43º = m*ax

Simplifying common terms, and solving for ax, we get:

ax = g*sin 43º - μk*g*cos 43º = 9.8 m/s²*sin 43º - 0.12*9.8 m/s²*cos 43º

ax= 5.8 m/s²