Starting with an initial speed v0, a mass m slides down a curved frictionless track (a shown in the figure below), arriving at the bottom with a speed v. From what vertical elevation h did it start?
To determine the vertical elevation h from which the mass m started, we can use the principle of conservation of energy. Since the track is frictionless, the only forces acting on the mass along the vertical direction are gravity and the normal force.
The initial kinetic energy of the mass m can be determined using the equation:
KE_initial = (1/2) * m * v0^2
The final kinetic energy at the bottom of the track can be calculated as:
KE_final = (1/2) * m * v^2
The potential energy at the starting point can be expressed as:
PE_initial = m * g * h
where g represents the acceleration due to gravity.
According to the conservation of energy principle, the initial kinetic energy plus the initial potential energy equals the final kinetic energy:
KE_initial + PE_initial = KE_final
Substituting the previously calculated values, we get:
(1/2) * m * v0^2 + m * g * h = (1/2) * m * v^2
Simplifying the equation:
(1/2) * v0^2 + g * h = (1/2) * v^2
Isolating h:
h = (1/2g) * (v^2 - v0^2)
Therefore, the vertical elevation from which the mass m started can be determined by using the equation h = (1/2g) * (v^2 - v0^2), where v is the final speed at the bottom of the track, v0 is the initial speed, and g is the acceleration due to gravity.
(PE+KE)initial = (PE+KE)final
mgh + mv0^2/2 = 0 + mv^2/2
=> h = (v^2-v0^2)/2g