a box slides down a smooth inclined plane 30m long and 10m high.determine the speed of the box at the bottom of the plane if it starts from rest at the top.
Vf^2 = Vo^2 + 2gd,
Vf^2 = 0 + 2*9.8*10 = 196,
Vf = 14m/s.
To determine the speed of the box at the bottom of the plane, we can use the principle of conservation of energy.
The potential energy at the top of the inclined plane (PE_top) is given by the equation:
PE_top = mgh
Where m is the mass of the box, g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the inclined plane (10m).
The kinetic energy at the bottom of the inclined plane (KE_bottom) is given by the equation:
KE_bottom = (1/2)mv^2
Where v is the velocity of the box at the bottom of the inclined plane.
According to the principle of conservation of energy, the total energy at the top of the inclined plane (PE_top) is equal to the total energy at the bottom of the inclined plane (KE_bottom). Therefore, we can equate the two equations:
mgh = (1/2)mv^2
Simplifying the equation, we can cancel out the mass of the box:
gh = (1/2)v^2
Now we can solve for v:
v^2 = 2gh
Taking the square root of both sides, we get:
v = √(2gh)
Plugging in the given values for g (9.8 m/s^2) and h (10m), we can calculate the velocity:
v = √(2 * 9.8 * 10) ≈ √(196) ≈ 14 m/s
Therefore, the speed of the box at the bottom of the plane is approximately 14 m/s.
To determine the speed of the box at the bottom of the inclined plane, we can use the principle of conservation of energy.
The initial potential energy of the box at the top of the plane is equal to the final kinetic energy of the box at the bottom of the plane. The formula to calculate potential energy is:
Potential Energy (PE) = mass (m) * acceleration due to gravity (g) * height (h)
The mass of the box is not provided, but it is not required to solve the problem because it cancels out when calculating the ratio of potential energy to kinetic energy. Therefore, we can ignore it in this calculation.
The acceleration due to gravity is approximately 9.8 m/s^2.
The initial potential energy of the box at the top is:
PE_initial = m * g * h
Here, h (height) is 10m.
The final kinetic energy of the box at the bottom is:
KE_final = 0.5 * m * v^2
Here, v is the final velocity of the box at the bottom.
According to the principle of conservation of energy:
PE_initial = KE_final
m * g * h = 0.5 * m * v^2
hence:
v^2 = 2 * g * h
Plugging in the values, we get:
v^2 = 2 * 9.8 m/s^2 * 10 m
v^2 = 196 m^2/s^2
Taking the square root of both sides, we get:
v = √196 m/s
v = 14 m/s
Therefore, the speed of the box at the bottom of the inclined plane is 14 m/s.