A cross-country skier slides horizontally along the snow and comes to rest after sliding a distance of 11 m. If the coefficient of kinetic friction between her skis and the snow is μk = 0.020, determine her speed at the start of the slide.

2.10m/s

Here, we use the work-energy principle that states that the work done (W) on a body is equal to the change in kinetic energy (Δ) of the body. i.e

W = Δ        [Δ = ₂ - ₁]

=> W = ₂ - ₁           ----------------(i)

[₂ = final kinetic energy ₁ = initial kinetic energy]

But;

W = F x s cos θ

Where;

F = net force acting on the body

s = displacement of the body due to the force

θ = angle between the force and the displacement.

Also;

₂ = x m x v²

₁ = x m x u²

Where;

m = mass of the body

v = final velocity of the body

u = initial velocity of the body

Substitute the values of ₂ , ₁ and W into equation (i) as follows;

F x s cos θ =  ( x m x v²) - ( x m x u²)    -----------------(ii)

From the question;

i. The skier comes to a rest, this implies that the final velocity (v) of the body(skier) is 0.

Therefore substitute v = 0 into equation (ii) to get;

F x s cos θ =  ( x m x 0²) - ( x m x u²)

F x s cos θ =  0 - ( x m x u²)

F x s cos θ =   - ( x m x u²)             ---------------------(iii)

ii. Since there is no motion in the vertical direction, the net force (F) acting is the kinetic frictional force () in the horizontal direction

i.e F =

But we know that the frictional force , is given by;

F = = μk x N

Where;

μk = coefficient of static friction

N = Normal reaction which is equal to the weight (m x g) of the skier [since there is no motion in the vertical]

=> F =    = μk x m x g          [m = mass of the skier and g = acceleration due to gravity]

iii. Also, since the only force acting is the frictional force acting to oppose motion, the angle θ between the force and the displacement is 180°

iv. Now substitute all of these values into equation (iii) as follows;

F x s cos θ =   - ( x m x u²)

μk x m x g x s cos θ = -  ( x m x u²)

Divide through by m;

μk x g x s cos θ = -  ( x u²)        ----------------(iv)

From the question;

s = 11m

μk = 0.020

Take g = 10m/s²

θ = 180°

Substitute these values into equation (iv) and solve for u;

0.020 x 10 x 11 cos 180 = -  ( x u²)

0.020 x 10 x 11 x (-1) = -  ( x u²)

-2.2 = -   x u²

u² = 4.4

u =

u = 2.10m/s

Therefore, the speed of the skier at the start of the slide is 2.10m/s

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