a 15kg crate initially at rest slides down a ramp 2m long and inclined at an angle of 20 degrees with the horizontal. If there is no friction between the ramp surface and crate, what is the velocity of the crate at the bottom of the ramp?
To find the velocity of the crate at the bottom of the ramp, we can use the principles of conservation of energy.
Step 1: Calculate the gravitational potential energy of the crate at the top of the ramp.
Gravitational potential energy (PE) is given by the formula:
PE = m * g * h
where m is the mass of the crate, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the vertical height.
Given:
m (mass of the crate) = 15 kg
h (vertical height) = 2 m
Using the formula:
PE = 15 kg * 9.8 m/s^2 * 2 m = 294 J
Step 2: Calculate the kinetic energy of the crate at the bottom of the ramp.
At the bottom of the ramp, all of the potential energy is converted into kinetic energy.
Kinetic energy (KE) is given by the formula:
KE = (1/2) * m * v^2
where v is the velocity of the crate at the bottom of the ramp.
Using the formula:
294 J = (1/2) * 15 kg * v^2
588 J = 15 kg * v^2
v^2 = 588 J / 15 kg ≈ 39.2 m^2/s^2
Step 3: Calculate the velocity of the crate at the bottom of the ramp.
Taking the square root of both sides:
v = √(39.2 m^2/s^2) ≈ 6.26 m/s
Therefore, the velocity of the crate at the bottom of the ramp is approximately 6.26 m/s.
To find the velocity of the crate at the bottom of the ramp, you can use the principle of conservation of energy. The potential energy at the top of the ramp is converted into kinetic energy at the bottom.
First, calculate the height of the ramp:
Given the angle of 20 degrees and the length of the ramp (2m), we can calculate the vertical height using trigonometry. The vertical height (h) can be found using the formula:
h = l * sin(angle)
where l is the length of the ramp and angle is the angle of inclination.
Substituting the values:
h = 2 * sin(20)
h ≈ 0.684 m
Now, calculate the potential energy (PE) of the crate at the top of the ramp:
PE = m * g * h
where m is the mass of the crate (15 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the ramp.
Substituting the values:
PE = 15 * 9.8 * 0.684
PE ≈ 101.646 J
Since there is no friction, the potential energy is completely converted into kinetic energy at the bottom of the ramp.
Kinetic energy (KE) can be calculated using the formula:
KE = (1/2) * m * v^2
where m is the mass of the crate and v is the velocity of the crate at the bottom of the ramp.
Setting the potential energy equal to the kinetic energy:
PE = KE
101.646 = (1/2) * 15 * v^2
Simplifying:
v^2 = (2 * 101.646) / 15
v^2 ≈ 13.5436
Taking the square root of both sides:
v ≈ 3.68 m/s
Therefore, the velocity of the crate at the bottom of the ramp is approximately 3.68 m/s.
h = 2*sin20 = 0.684 m.
V^2 = Vo^2 + 2g*h.
Vo = 0.