A 55.3 kg skier is moving at 14.9 m/s on a frictionless horizontal, snow covered plateau when she encounters a rough patch 1.7 m long. The coefficient of kinetic friction between this patch and her skis is 0.27. After crossing the rough patch, what is the speed of the skier in ms-1 ? g=9.8ms-2
KE initial- Work lost to Friction = KE final
Ff = mu mg = .27*55.3*9.8
so
1/2 55.3 14.9^2 - Ff(1.7) = 1/2 55.3 vf^2
solve for vf
thanks:)
To find the final speed of the skier after crossing the rough patch, we need to consider the work done by the friction force.
First, let's calculate the work done by the friction force. The work done by a force is given by the formula:
Work = Force x Distance x Cosine(angle)
In this case, the force is the friction force (Ffriction), the distance is the length of the rough patch (1.7 m), and the angle between the force and the direction of motion is 180 degrees (as the force is opposing the skier's motion). The formula simplifies to:
Work = -Ffriction x Distance
Next, let's calculate the friction force (Ffriction). The friction force can be calculated using the formula:
Ffriction = μ x Normal force
The normal force (N) is the force exerted by the surface on the skier and is equal to the skier's weight (mg) in this case. The formula simplifies to:
Ffriction = μ x mg
Substituting this into the work equation, we have:
Work = -μ x mg x Distance
The work done by the friction force is equal to the change in kinetic energy (ΔKE) of the skier. The change in kinetic energy can be calculated using the formula:
ΔKE = 0.5 x m x (vf^2 - vi^2)
Where m is the mass of the skier, vf is the final velocity, and vi is the initial velocity. Rearranging the equation to solve for the final velocity, we get:
vf^2 = (2 x ΔKE / m) + vi^2
Since the initial velocity (vi) is given as 14.9 m/s and the change in kinetic energy (ΔKE) is the work done by the friction force, we can write:
vf^2 = (2 x |Work| / m) + vi^2
Now, let's substitute the given values into the equations and solve for the final velocity (vf):
Mass of skier (m) = 55.3 kg
Initial velocity (vi) = 14.9 m/s
Distance (d) = 1.7 m
Coefficient of kinetic friction (μ) = 0.27
Acceleration due to gravity (g) = 9.8 m/s^2
First, let's calculate the friction force (Ffriction):
Ffriction = μ x mg
= 0.27 x 55.3 kg x 9.8 m/s^2
= 144.9999 N (approximately 145 N)
Next, let's calculate the work done by the friction force (Work):
Work = -Ffriction x Distance
= -145 N x 1.7 m
= -246.5 J
Now, let's calculate the final velocity (vf):
vf^2 = (2 x |Work| / m) + vi^2
= (2 x 246.5 J / 55.3 kg) + (14.9 m/s)^2
= 28.1799... + 222.01
= 250.1899... m^2/s^2
Taking the square root of both sides to solve for vf:
vf = sqrt(250.1899... m^2/s^2)
= 15.8191... m/s
Therefore, the final speed of the skier after crossing the rough patch is approximately 15.8 m/s.