A woman is riding a Jet Ski at a speed of 20 m/s and notices a seawall straight ahead. The farthest she can lean the craft in order to make a turn is 20°. This situation is like that of a car on a curve that is banked at an angle of 20°. 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 formula for the radius of a curved motion.
Let's start by calculating the radius of the turn. The centripetal force is provided by the horizontal component of the water's reaction force acting on the Jet Ski. This force can be calculated using the formula:
F = m * a
where,
F = centripetal force
m = mass of the Jet Ski
a = centripetal acceleration
The centripetal acceleration can be calculated using the formula:
a = v^2 / r
where,
v = velocity of the Jet Ski
r = radius of the turn
Now, since the woman is trying to make the turn without slowing down, the force of friction between the Jet Ski and the water provides the centripetal force. The frictional force can be calculated using the formula:
f = μ * N
where,
f = frictional force
μ = coefficient of friction
N = normal force
The normal force N is equal to the weight of the Jet Ski, which can be calculated using the formula:
N = m * g
where,
g = acceleration due to gravity
Now, to find the radius of the turn, we can equate the centripetal force and the frictional force:
f = F
μ * N = m * a
μ * m * g = m * (v^2 / r)
Simplifying the equation, we get:
r = g / (μ * (v^2 / g))
r = g^2 / (μ * v^2)
Now, given that the angle of bank is 20°, we can calculate the coefficient of friction (μ) using the following formula:
μ = tan(20°)
Once we have the value of μ, we can substitute it into the equation for the radius to find the minimum distance from the seawall.