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 find the time it takes for the ice skater's velocity to be reduced to +2.26 m/s, we need to use the concept of kinetic friction.
The force of kinetic friction can be calculated using the equation:
Fk = μk * N
where Fk is the force of kinetic friction, μk is the coefficient of kinetic friction, and N is the normal force.
In this case, the normal force is equal to the weight of the skater, which is given by:
N = m * g
where m is the mass of the skater and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Since the skater is gliding horizontally, the force of kinetic friction acts as the only force opposing her motion, causing her velocity to decrease.
Using Newton's second law, we can express the force of kinetic friction as:
Fk = m * a
where a is the acceleration of the skater.
Since we know that force of kinetic friction is equal to μk * N, and force of kinetic friction is equal to m * a, we can set the two equations equal to each other:
μk * N = m * a
Substituting N = m * g, we get:
μk * m * g = m * a
Simplifying the equation by dividing both sides by m, we have:
μk * g = a
This tells us that the acceleration of the skater is equal to μk * g.
Now, we can use the basic kinematic equation to find the time it takes for the skater's velocity to be reduced to +2.26 m/s:
v = u + a * t
where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time.
Rearranging the formula, we have:
t = (v - u) / a
Substituting the given values into the equation, we get:
t = (2.26 - 5.33) / (μk * g)
Substituting the given value of the coefficient of kinetic friction (μk = 0.0988) and the acceleration due to gravity (g = 9.8 m/s^2), we can calculate the time it takes for the skater's velocity to be reduced to +2.26 m/s.