A 61 - kg ice skater coasts with no effort for 61 m until she stops. If the coefficient of kinetic friction between her skates and the ice is μk=0.10, how fast was she moving at the start of her coast?

To find the speed at the start of the coast, we can use the concept of work and energy.

The work done by friction can be calculated using the formula:
Work = Force * Distance

In this case, the force of friction can be found using the formula:
Frictional Force = μk * Normal Force

The normal force is equal to the weight of the ice skater, which can be calculated using:
Normal Force = mass * gravity

The work done by friction can then be expressed as:
Work = μk * Normal Force * Distance

The work done by friction is equal to the change in kinetic energy of the skater:
Work = ΔKE

Since the skater comes to a stop, her initial kinetic energy is equal to the work done by friction:
Initial KE = Work = μk * Normal Force * Distance

The initial kinetic energy can be expressed as:
Initial KE = (1/2) * mass * initial velocity^2

Rearranging the equation to solve for the initial velocity, we get:
initial velocity = sqrt(2 * (μk * Normal Force * Distance) / mass)

Now, we can substitute the given values and calculate the initial velocity:
mass = 61 kg
Distance = 61 m
μk = 0.10
gravity = 9.8 m/s^2

Normal Force = mass * gravity
Normal Force = 61 kg * 9.8 m/s^2 = 598.8 N

initial velocity = sqrt(2 * (0.10 * 598.8 N * 61 m) / 61 kg)
initial velocity = sqrt(1200 N * m / 61 kg)
initial velocity = sqrt(19.6721 m^2/s^2)
initial velocity = 4.43 m/s

Therefore, the skater was moving at an initial speed of 4.43 m/s before her coast.

To find the initial velocity of the ice skater, we can use the principle of energy conservation. The initial kinetic energy of the skater will be equal to the work done by the friction force in bringing the skater to a stop.

The work done by the friction force can be calculated using the formula: Work = Force × Distance × cosθ. In this case, the force is the friction force, the distance is the distance traveled by the skater, and θ is the angle between the force and the direction of motion (which is 0 degrees because the force of friction is opposing the motion).

The work done by friction can be determined as follows: Work = μk × m × g × d, where μk is the coefficient of kinetic friction, m is the mass of the skater, g is the acceleration due to gravity (approximately 9.8 m/s^2), and d is the distance traveled by the skater.

We now have the equation: Initial Kinetic Energy = Work.
The initial kinetic energy of the skater is given by the formula: (1/2) × m × v^2, where m is the mass of the skater and v is the initial velocity.

Equating the two equations, we get:
(1/2) × m × v^2 = μk × m × g × d

Simplifying the equation, we can cancel out the mass:
(1/2) × v^2 = μk × g × d

Rearranging the equation to solve for v:
v^2 = 2 × μk × g × d

Taking the square root of both sides, we find:
v = √(2 × μk × g × d)

Now we can substitute the given values:
μk = 0.10 (coefficient of kinetic friction)
m = 61 kg (mass of the skater)
g = 9.8 m/s^2 (acceleration due to gravity)
d = 61 m (distance traveled by the skater)

Plugging in these values, we get:
v = √(2 × 0.10 × 9.8 × 61)

Calculating this expression, we find that the initial velocity of the skater was approximately 15.29 m/s.

vf^2=vi^2+2ad

and a=force/mass=mu*mg/m=mu*g
solve for vi