A)determine the velocity v as a function of the vertical height H b)How does the velocity v depend upon m?

Question

A block of mass m slides from rest down a smooth frictionless incline at an angle to the horizontal

Answer 1

This is a simple energy question.

Change in potential energy = change in kinetic energy, where velocity is the kinetic and gravitational energy is the height part.

SO for a)

delta P.E. = delta K.E.

m * g * H = 1/2 * m * v^2

i.e.  gH = 1/2 v^2

and for b) the velocity will not change for a varying m, but it will change for a varying H, because of the relationship between kinetic and potential energy

Answer 2

s = 1/2at^2

H-s= H-1/2at^2

s is position, or distance traveled.

H is initial height.

a is acceleration, which is some fraction of gravity, depending on the angle of the incline.

v does not depend on mass

Answer 3

A:

The acceleration is the sine of the angle of depression of the block multiplied by gravity (~9.8 on the Earth's surface). The velocity v = a (as defined above) * t (time in seconds).

If we let i = initial height,

the current height H = i - (a * t^2 / 2).

So a * t^2 = 2 * (i - H).

Therefore t = sqrt(2 * (i - H) / a). If we substitute that into v = a*t, v = a * sqrt (2* (i-H)/a). Note that i and a are constants, so the only independent variable is the current height H.

B:

As for the mass m, the acceleration does not change based on m. Think back to the story of Galileo Galilei dropping weights off the Leaning Tower of Pisa. Without friction of any sort, any two objects of different mass will fall at the same rate (except, if the mass of the object was somehow to equal the mass of the earth, there would be no acceleration due to gravity, as the object would pull equally on the earth as the earth pulls on it).