A woman is riding a Jet Ski at a speed of 23 m/s and notices a seawall straight ahead. The farthest she can lean the craft in order to make a turn is 26°. This situation is like that of a car on a curve that is banked at an angle of 26°. If she tries to make the turn without slowing down, what is the minimum distance from the seawall that she can begin making her turn and still avoid a crash?
To determine the minimum distance from the seawall that the woman can begin making her turn and still avoid a crash, we need to consider the concept of centripetal force.
Centripetal force is the force that keeps an object moving in a curved path and is directed towards the center of the curve. It is given by the equation:
F = (mv²) / r
where F is the centripetal force, m is the mass of the object, v is the velocity, and r is the radius of the curved path.
In this case, the woman riding the Jet Ski is like a car on a banked curve. The angle of the seawall (26°) represents the banking angle.
When a car or any other vehicle is on a banked curve, the frictional force between the tires and the road provides the necessary component of centripetal force to keep the vehicle moving in a curved path. This means that the gravitational force (mg) acting on the vehicle is not the only force involved in the turn.
The component of the gravitational force acting parallel to the surface of the road can be represented as mg * sinθ, where θ is the angle of the banking. This component of the force acts towards the center of the curve.
So, in this case, the centripetal force is the sum of the component of the gravitational force (mg * sinθ) and the frictional force (μmg * cosθ), where μ is the coefficient of friction between the Jet Ski and the water.
Let's assume the mass of the Jet Ski is m, and the radius of the curved path is r (the minimum distance from the seawall that the woman can begin making her turn).
The centripetal force can be written as:
F = mg * sinθ + μmg * cosθ
Since the speed of the Jet Ski is given as 23 m/s, we can calculate the velocity using the equation v = ωr, where ω is the angular velocity.
The angular velocity can be calculated as ω = v / r.
Now we can substitute ω into the centripetal force equation, which gives us:
(mg * sinθ + μmg * cosθ) = (mv²) / r
Since we are interested in finding the minimum distance from the seawall (r), we can solve this equation for r.
r = (mv²) / (mg * sinθ + μmg * cosθ)
Now we can substitute the given values into the equation and calculate the minimum distance from the seawall.