A woman is riding a Jet Ski at a speed of 27.9 m/s and notices a seawall straight ahead. The farthest she can lean the craft in order to make a turn is 21.0°. This situation is like that of a car on a curve that is banked at an angle of 21.0°. 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 find the minimum distance from the seawall that the woman can begin making her turn and still avoid a crash, we can use the concept of centripetal force and the angle of banking.
1. Determine the acceleration needed:
- The centripetal force required to keep the woman moving in a curved path is given by the formula: F = (mv²) / r, where F is the centripetal force, m is the mass of the Jet Ski, v is the velocity, and r is the radius of the turn.
- The gravitational force acting on the Jet Ski can be resolved into two components: perpendicular to the slope of the seawall (Fn - normal force) and parallel to the slope of the seawall (Fg - gravitational force).
- The angle of banking (θ) and the component of gravitational force (Fg) in the direction of the normal force (Fn) are related by Fg = Fn * tan(θ).
- Since the normal force (Fn) is equal and opposite to the centripetal force (F), we have: F = Fg = Fn * tan(θ).
- Substituting the expression for centripetal force, we get: (mv²) / r = Fn * tan(θ).
- Rearranging the equation, we find: r = (m * v²) / (Fn * tan(θ)), where r is the radius of the turn.
- Further rearranging, we get: r = (m * v²) / (m * g * tan(θ)), where g is the acceleration due to gravity.
2. Determine the minimum distance:
- To find the minimum distance, we need to calculate the radius (r) of the turn.
- Once we have the radius, we can consider it as the hypotenuse of a right triangle, with the horizontal distance from the seawall as the adjacent side.
- The vertical distance from the seawall to the starting point of the turn is given by r * sin(θ).
- Hence, the minimum distance is the horizontal distance subtracted by the vertical distance: d = r - r * sin(θ).
3. Substituting the given values:
- The speed of the Jet Ski is given as 27.9 m/s.
- The angle of banking is given as 21.0°.
Let's calculate the minimum distance step by step.
To calculate the minimum distance from the seawall that the woman can begin making her turn and avoid a crash, we need to consider the centripetal force acting on the Jet Ski.
The formula for the centripetal force is given by:
F = m * v^2 / r
Where:
F = centripetal force
m = mass of the object (in this case the Jet Ski)
v = velocity of the object
r = radius of the turn
In this case, the centripetal force is provided by the banking angle of the seawall, which is 21.0°. The formula for the banking angle is:
tan(θ) = v^2 / (g * r)
Where:
θ = banking angle
v = velocity of the object
g = acceleration due to gravity (approximately 9.8 m/s^2)
r = radius of the turn
We can solve this equation for r:
r = v^2 / (g * tan(θ))
Now, we can substitute the given values into the equation:
v = 27.9 m/s
θ = 21.0° (converted to radians by multiplying by π/180)
g = 9.8 m/s^2
r = (27.9)^2 / (9.8 * tan(21.0°))
Using a scientific calculator or an online trigonometric calculator, we can find the value of tan(21.0°), which is approximately 0.381.
r = (27.9)^2 / (9.8 * 0.381)
Calculating this, we get:
r ≈ 216.997 m
So, the minimum distance from the seawall that the woman can begin making her turn and still avoid a crash is approximately 216.997 meters.