An ice skater slides (without pushing) across a frozen pond for 25.0 m before coming to a stop. If the coefficient of kinetic friction is 0.050, how fast was the skater initially moving?
m•v²/2 =μ•m•g•s
v=sqrt(2• μ•g•s)
To find the initial speed of the ice skater, we can use the concept of work and energy. The work done by the friction force is equal to the change in kinetic energy of the skater.
The work done by friction can be calculated using the formula:
Work = Force x Distance x cos(θ)
In this case, the force is the friction force, distance is the distance traveled by the skater, and θ is the angle between the force and the displacement (which is 0 degrees since the force is opposite to the displacement).
The friction force can be calculated using the formula:
Friction Force = Coefficient of Friction x Normal Force
The normal force is the force exerted on the skater perpendicular to the surface, which is equal to the weight of the skater (mg) since there is no vertical acceleration.
Combining the formulas, we get:
Work = (Coefficient of Friction x Normal Force) x Distance
Since the work done by friction is equal to the change in kinetic energy, we can write:
0.5 x m x (v^2 - 0) = (Coefficient of Friction x m x g) x Distance
Where:
m is the mass of the skater
v is the initial speed of the skater
g is the acceleration due to gravity (9.8 m/s^2)
Simplifying the equation, we get:
0.5 x v^2 = Coefficient of Friction x g x Distance
Now we can substitute the given values:
Coefficient of Friction = 0.050
g = 9.8 m/s^2
Distance = 25.0 m
Plugging in these values, we can solve for v:
0.5 x v^2 = 0.050 x 9.8 x 25.0
0.5 x v^2 = 12.25
v^2 = 12.25 / 0.5
v^2 = 24.5
v = √(24.5)
v ≈ 4.95 m/s
Therefore, the initial speed of the skater was approximately 4.95 m/s.