A mass of 18 kg is being pushed up a frictionless incline with a constant force of 20 Newtons directed parallel to the incline, up the incline. At the top of the incline the mass is moving at 23 m/s up the incline. If the angle of inclination is 58 degrees and the height (not length) of the incline is 24 meters, what was the magnitude of the mass's velocity at the bottom of the incline in m/s?
Vf^2 = Vo^2 + 2g*d,
Vo^2 = Vf^2 - 2g*d,
Vo^2 -= (23)^2 - (-19.6)*24 = 999.4
Vo = 3i.6m/s = Yo = ver. component of
initial velocity @ bottom of incline.
Vo = Yo / sinA = 31.6 / sin58=37.3m/s.
= Velocity at bottom of incline.
To find the magnitude of the mass's velocity at the bottom of the incline, we can use the principle of conservation of mechanical energy. The mechanical energy is conserved when there is no friction, so the initial mechanical energy is equal to the final mechanical energy.
The initial mechanical energy is the potential energy at the top of the incline, which is given by the formula: potential energy = mass * gravity * height * sin(theta)
Here, the mass is 18 kg, gravity is 9.8 m/s^2, height is 24 meters, and theta is the angle of inclination which is 58 degrees.
Initial potential energy = 18 kg * 9.8 m/s^2 * 24 meters * sin(58 degrees)
Next, let's find the final mechanical energy, which is the kinetic energy at the bottom of the incline. The kinetic energy is given by the formula: kinetic energy = 1/2 * mass * velocity^2.
We need to find the velocity at the bottom of the incline. To do this, we can use Newton's second law (F = ma) and apply it to the mass's motion up the incline.
The net force acting on the mass is the force applied parallel to the incline minus the force due to gravity. Since the incline is frictionless, there is no force due to friction.
Net force = applied force - force due to gravity
The applied force is given as 20 Newtons, and the force due to gravity is given as mass * gravity * cos(theta).
Net force = 20 Newtons - (18 kg * 9.8 m/s^2 * cos(58 degrees))
Now, we can find the acceleration using Newton's second law:
acceleration = net force / mass
Next, we can use the kinematic equation to find the final velocity:
final velocity^2 = initial velocity^2 + 2 * acceleration * distance
Given that the initial velocity is 23 m/s up the incline, the distance is the height of the incline which is 24 meters, and the acceleration is the value we calculated.
Finally, we can solve for the final velocity by taking the square root of both sides of the equation.
Now we have the final velocity at the bottom of the incline, which is the magnitude of the mass's velocity.