An ice skater is gliding horizontally across the ice with an initial velocity of +5.33 m/s. The coefficient of kinetic friction between the ice and the skate blades is 0.0988, and air resistance is negligible. How much time elapses before her velocity is reduced to +2.26 m/s?
To determine the time it takes for the ice skater's velocity to be reduced to +2.26 m/s, we need to calculate the acceleration first. Let's proceed step by step:
Step 1: Identify the given values:
- Initial velocity, u = +5.33 m/s
- Final velocity, v = +2.26 m/s
- Coefficient of kinetic friction, μ = 0.0988
Step 2: Calculate the acceleration:
The force of friction can be calculated using the equation:
F_friction = μ * Normal force
The normal force is equal to the weight of the skater, which can be calculated using:
Weight = mass * acceleration due to gravity
Since the skater is gliding horizontally, the normal force is equal to the weight:
Normal force = Weight = mass * acceleration due to gravity
The net force on the skater is given by:
Net force = Force of friction = mass * acceleration
Using Newton's second law, we can say that:
Net force = mass * acceleration
Equating the above two equations, we get:
mass * acceleration = μ * (mass * acceleration due to gravity)
Canceling out the mass on both sides, we get:
acceleration = μ * acceleration due to gravity
Substituting the values, we have:
acceleration = 0.0988 * 9.8 m/s^2 ≈ 0.96944 m/s^2
Step 3: Calculate the time taken:
We can use the equation of motion to calculate the time taken to change from the initial velocity to the final velocity:
v = u + a * t
Rearranging the equation, we have:
t = (v - u) / a
Substituting the values, we get:
t = (2.26 m/s - 5.33 m/s) / -0.96944 m/s^2
Calculating the expression, we find:
t ≈ -3.07 s
However, time cannot be negative in this context. This negative sign indicates that the velocity is decreasing. Taking the absolute value, we find:
t ≈ 3.07 s
Therefore, it takes approximately 3.07 seconds for the skater's velocity to be reduced to +2.26 m/s.
To find the time it takes for the skater's velocity to be reduced to +2.26 m/s, we can use the concept of kinetic friction.
The equation for kinetic friction is given by:
friction = coefficient of kinetic friction * normal force
The normal force is the force exerted by a surface perpendicular to the object. Since the skater is gliding horizontally, the normal force is equal to the weight of the skater.
Now, let's break down the problem into steps:
Step 1: Find the weight of the skater.
The weight of an object is given by the equation:
weight = mass * gravitational acceleration
In this case, the weight is the normal force since the skater is on a horizontal surface. The mass of the skater is not given, so we can skip this step for now.
Step 2: Calculate the friction force.
The friction force is given by the equation:
friction = coefficient of kinetic friction * normal force
In this case, the normal force is equal to the weight of the skater.
Step 3: Calculate the acceleration.
The net force acting on the skater is the force of kinetic friction.
According to Newton's second law:
net force = mass * acceleration
The net force can be considered as the friction force in this case.
Step 4: Calculate the time taken.
We can use the following kinematic equation to find the time taken:
velocity = initial velocity + (acceleration * time)
Rearranging the equation:
time = (velocity - initial velocity) / acceleration
Now, let's calculate the time taken.
Given:
Initial velocity (v_0) = +5.33 m/s
Final velocity (v) = +2.26 m/s
Coefficient of kinetic friction (μ) = 0.0988
Step 1: Calculate the friction force.
We can calculate the friction force by multiplying the coefficient of kinetic friction (μ) by the normal force.
Friction = μ * Normal force
Step 2: Calculate the acceleration.
Since the net force is equal to the friction force, we can calculate the acceleration by:
Acceleration = Friction / mass
Step 3: Calculate the time taken.
Plug in the values of the initial velocity (v_0), final velocity (v), and acceleration into the equation:
time = (v - v_0) / acceleration
Using these steps and the given values, we can determine the time it takes for the skater's velocity to decrease to +2.26 m/s.