1. An object of mass m and initial speed v0 sliding on a horizontal frictionless surface encounters a frictionless ramp. To what vertical height will the object rise before coming to rest?
To determine the vertical height the object will rise before coming to rest, we'll need to apply the principle of conservation of mechanical energy. The initial kinetic energy of the object will be entirely converted into gravitational potential energy at the maximum height reached.
1. Firstly, let's consider the forces acting on the object. Since there is no friction, the only force acting on the object is its weight (mg), where m is the mass of the object and g is the acceleration due to gravity.
2. Next, we need to determine the initial kinetic energy (K) of the object. The kinetic energy is given by the formula K = (1/2)mv0^2, where m is the mass of the object and v0 is the initial speed.
3. As the object slides up the frictionless ramp, its kinetic energy will gradually convert into gravitational potential energy (U). The potential energy is given by the formula U = mgh, where m is the mass of the object, g is the acceleration due to gravity, and h is the height.
4. At the maximum height reached, the object comes to rest, meaning its final velocity (vf) is zero. Therefore, the final kinetic energy (Kf) will be zero.
5. Applying the conservation of mechanical energy, we equate the initial kinetic energy to the final potential energy: K = U.
(1/2)mv0^2 = mgh
6. Solving for h, we get: h = (v0^2)/(2g).
Therefore, the vertical height the object will rise before coming to rest is given by the equation h = (v0^2)/(2g), where v0 is the initial speed of the object and g is the acceleration due to gravity.