Determine the coefficient of friction between a girl's skis and the snow when she skis down a 20° degree hill and her ending speed is exactly two-fifths of what it would have been if the slope had been frictionless.
two-fifths speed, that means KE is 4/25 of all the energy.
find her speed if no friction
mgh=1/2 mv^2
v= sqrt(2gh)
her speed is 2/5 of this , or energy is 4/25 of original energy.
friction energy=mgh(4/25)
mu*cos20*mg*h/sin20=mgh(4/25)
mu=tan20*(4/25) check my work
To determine the coefficient of friction between the girl's skis and the snow, we can use the concept of work and energy. The work done by friction is equal to the change in mechanical energy of the system.
Let's break down the problem step by step:
Step 1: Determine the mechanical energy of the system on the frictionless slope.
Since the slope is frictionless, the only forces acting on the girl are gravity and the normal force. The work done by gravity is given by the formula: work = mgh, where m is the mass of the girl, g is the acceleration due to gravity, and h is the vertical height of the hill.
However, on a frictionless slope, there is no work done by friction, so the work done by gravity equals the change in mechanical energy of the system.
Step 2: Calculate the change in mechanical energy on the frictionless slope.
The initial mechanical energy is given by: E_initial = mgh, where m is the mass of the girl, g is the acceleration due to gravity, and h is the height of the hill.
The final mechanical energy is given by: E_final = (1/2)mv^2, where m is the mass of the girl, and v is the speed of the girl at the bottom of the hill.
Step 3: Determine the mechanical energy of the system on the actual slope.
Since the actual slope has a coefficient of friction, we need to consider the work done by friction as well. The mechanical energy is given by the same formula as the frictionless slope: E_actual = (1/2)mv_final^2.
Step 4: Calculate the work done by friction.
The work done by friction is given by: work_friction = E_initial - E_actual.
Step 5: Determine the coefficient of friction.
The work done by friction can also be expressed as: work_friction = friction_force * distance, where friction_force is the force of friction and distance is the distance traveled down the slope. The force of friction can be calculated by: friction_force = coefficient_of_friction * normal_force. The normal force is equal to the girl's weight, which is given by: normal_force = m * g.
Substituting these values into the equation above, we get: work_friction = coefficient_of_friction * m * g * distance.
Now, set the equations for the work done by friction equal to each other:
E_initial - E_actual = coefficient_of_friction * m * g * distance.
Rearrange the equation to solve for the coefficient of friction:
coefficient_of_friction = (E_initial - E_actual) / (m * g * distance).
Plug in the given values, and you can now determine the coefficient of friction between the girl's skis and the snow when she skis down the 20° degree hill with her ending speed at two-fifths of the frictionless slope's speed.