A skier of mass 77.7 kg, starting from rest, slides down a slope at an angle $\theta$ of 35.7° with the horizontal. The coefficient of kinetic friction is 0.08. What is the net work, i.e. net gain in kinetic energy, (in J) done on the skier in the first 10.9 s of descent?
Ws = m*g = 77.7kg * 9.8N/kg = 761.5 N = Wt. of skier.
Fp = 761.5*sin35.7 = 444.3 N. = Force parallel to slope.
Fn = 761.5*cos35.7 = 618.4 = Normal = Force perpendicular to slope.
Fk = u*Fn = 0.08 * 618.4 = 49.47 N. =
Force of kinetic friction.
Fp-Fk = m*a
a=(Fp-Fk)/m=(444.3-49.47)/77.7=5.08m/s^2
V = Vo + a*t
V = 0 + 5.08*10.9 = 55.39 m/s.
Work=m*V^2/2 = 77.7*55.39^2/2=119,194 J.
To find the net work done on the skier in the first 10.9 seconds of descent, we need to calculate the change in kinetic energy during that time.
The net work done on an object is given by the equation:
Net work = Change in kinetic energy
The change in kinetic energy can be calculated as the difference between the final kinetic energy and the initial kinetic energy:
Change in kinetic energy = Final kinetic energy - Initial kinetic energy
Since the skier is starting from rest, the initial kinetic energy is zero. Therefore, we only need to find the final kinetic energy.
The final kinetic energy can be calculated using the formula:
Final kinetic energy = (1/2) * mass * velocity^2
To determine the velocity of the skier, we can use the concepts of Newton's second law and the forces acting on the skier.
The force parallel to the slope acting on the skier can be calculated using the equation:
Force parallel = (mass * gravity * sin(theta)) - (coefficient of friction * mass * gravity * cos(theta))
where:
- mass is the mass of the skier (77.7 kg)
- gravity is the acceleration due to gravity (9.8 m/s^2)
- theta is the angle of the slope (35.7°)
- coefficient of friction is 0.08
Next, we can use Newton's second law to determine the acceleration of the skier along the slope:
Force parallel = mass * acceleration
Rearranging the equation, we get:
acceleration = Force parallel / mass
Once we have the acceleration, we can calculate the final velocity using the equation of motion:
velocity = initial velocity + (acceleration * time)
Since the skier starts from rest, the initial velocity is zero.
Now, we have the final velocity, and we can calculate the final kinetic energy using the formula mentioned earlier.
Finally, we can substitute the values into the equation for net work to find the answer.