A 60 kg skier with an initial velocity of 12 m/s coasts up a hill. At the top of the hill, the skier is traveling at 9.75 m/s. Assuming there is no friction in the skis, how high was the hill?
To solve this problem, we can use the principle of conservation of mechanical energy. At the top of the hill, all the initial kinetic energy of the skier is transformed into potential energy.
The initial kinetic energy of the skier is given by:
KE_initial = (1/2) * m * v_initial^2
where m is the mass of the skier (60 kg) and v_initial is the initial velocity (12 m/s).
Plugging in the values, we have:
KE_initial = (1/2) * 60 kg * (12 m/s)^2
= 4320 J
At the top of the hill, the skier has only potential energy, given by:
PE_top = m * g * h
where g is the acceleration due to gravity (9.8 m/s^2) and h is the height of the hill.
Since the mechanical energy is conserved, we have:
KE_initial = PE_top
4320 J = 60 kg * 9.8 m/s^2 * h
Rearranging the equation, we can solve for h:
h = 4320 J / (60 kg * 9.8 m/s^2)
= 7 m
Therefore, the height of the hill is 7 meters.