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?
A mass of 5.6 kg lies at the top of a frictionless incline. It is inclined at an angle such that the normal force on the mass is 30 Newtons. If the length of the incline is 58 meters and the mass is originally at rest at the top of the incline and then released, how long does it take for the mass to reach the bottom of the incline in seconds?
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 at the top of the incline, when the mass is moving at 23 m/s, is given by the sum of its potential energy and kinetic energy.
The potential energy (PE) of an object at a certain height is given by the equation:
PE = m * g * h
where m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height.
In this case, the height of the incline is given as 24 meters, so the potential energy at the top is:
PE_top = 18 kg * 9.8 m/s² * 24 m
Next, we find the kinetic energy (KE) at the top of the incline using the equation:
KE = (1/2) * m * v²
where v is the velocity.
In this case, the velocity at the top of the incline is 23 m/s, so the kinetic energy at the top is:
KE_top = (1/2) * 18 kg * (23 m/s)²
The total mechanical energy (TE) at the top of the incline is the sum of the potential energy and kinetic energy:
TE_top = PE_top + KE_top
According to the principle of conservation of mechanical energy, the total mechanical energy at the top of the incline should be equal to the total mechanical energy at the bottom of the incline (assuming no energy losses due to friction or other factors).
At the bottom of the incline, the height is 0, so the potential energy is also 0:
PE_bottom = 0
Since there is no information given about the velocity at the bottom of the incline, we can represent it as v_bottom.
The kinetic energy at the bottom is given by: KE_bottom = (1/2) * m * (v_bottom)²
Since there is no friction mentioned, the total mechanical energy at the bottom of the incline should be equal to the total mechanical energy at the top:
TE_top = TE_bottom
Using the equations above, we can write the equation:
PE_top + KE_top = PE_bottom + KE_bottom
Now, let's substitute the known values into the equation and solve for v_bottom.
PE_top = 18 kg * 9.8 m/s² * 24 m
KE_top = (1/2) * 18 kg * (23 m/s)²
PE_bottom = 0
KE_bottom = (1/2) * 18 kg * (v_bottom)²
Substituting these values into our equation, we get:
18 kg * 9.8 m/s² * 24 m + (1/2) * 18 kg * (23 m/s)² = 0 + (1/2) * 18 kg * (v_bottom)²
Now we can solve for v_bottom.