Allison is skateboarding across a level road. The coefficient of static friction is 0.680 between Allison's shoes and the skateboard. If there is a hole in the road 23.2 m up ahead and Allison is able to stop without sliding off of the skateboard, what is the maximum velocity she could have initially been travelling?
To determine the maximum velocity Allison could have initially been traveling, we need to consider the forces at play and the condition for stopping without sliding off the skateboard.
First, let's look at the forces acting on Allison as she is skateboarding:
1. Weight (mg) acts vertically downwards.
2. Normal force (N) acts upwards perpendicular to the road.
3. Frictional force (f) opposes the motion.
In this case, it is important to note that the maximum force of static friction (fs), given by fs = μsN, is acting on Allison to prevent her from sliding off the skateboard.
To calculate the maximum velocity, we need to determine the maximum frictional force and equate it to the net force (ma) required to stop Allison. Assuming Allison's mass is 'm', we have the following equation:
fs = μsN = ma
Since the normal force and weight cancel each other out, we have:
μsN = ma
Next, let's consider the condition for stopping without sliding. When Allison reaches the hole in the road, she must be able to stop within the given distance of 23.2 m. The stopping distance (d) can be calculated using the following equation:
d = (v^2 - u^2) / (2a)
where:
- v is the final velocity (which is zero)
- u is the initial velocity
- a is the deceleration due to the maximum static friction force
Substituting values, we have:
23.2 = (0 - u^2) / (2a)
Simplifying the equation, we get:
u^2 = 2ad
To find the maximum velocity (u) Allison could have initially been traveling, we need to solve for u:
u = √(2ad)
Plug in the given value of the stopping distance (d = 23.2 m) and the coefficient of static friction (μs = 0.680) to get:
u = √(2 * 0.680 * a * 23.2)
Now, we need to determine the maximum deceleration (a) that Allison experiences when she stops. This occurs when the maximum force of static friction (fs) is equal to the net force acting on Allison.
The net force can be calculated using Newton's second law, F = ma. In this case, the net force will be due to the force of static friction:
fs = ma
Substituting the value of fs (μsN) into the equation, we have:
μsN = ma
Since N = mg (weight), we can rewrite the equation as:
μs(mg) = ma
Simplifying, we get:
μsg = a
Now we have the value of the maximum deceleration (a = μsg), which we can substitute back into the equation to find the maximum velocity:
u = √(2 * 0.680 * μsg * 23.2)
Plug in the value of g (acceleration due to gravity) and μs to calculate the maximum velocity:
u = √(2 * 0.680 * 9.8 * 23.2)
Calculating this expression will give us the maximum velocity that Allison could have initially been traveling.