First, we need to find the final velocity of the brick using the kinematic equation:
vf^2 = vi^2 + 2ad
where vf is the final velocity, vi is the initial velocity (7.50 m/s), a is the acceleration, and d is the distance (2.60 m).
The acceleration can be found using the component of the force of gravity that is parallel to the inclined plane:
a = gsinθ
where g is the acceleration due to gravity (9.81 m/s^2) and θ is the angle of the inclined plane (29.0o).
a = (9.81 m/s^2)sin(29.0o) = 4.74 m/s^2
Substituting the values into the kinematic equation, we get:
vf^2 = (7.50 m/s)^2 + 2(4.74 m/s^2)(2.60 m)
vf^2 = 78.61 m^2/s^2
vf = 8.87 m/s
(i) The change in kinetic energy of the brick is:
ΔK = 1/2mvf^2 - 1/2mvi^2
where m is the mass of the brick (4.0 kg).
ΔK = 1/2(4.0 kg)(8.87 m/s)^2 - 1/2(4.0 kg)(7.50 m/s)^2
ΔK = 60.1 J
(ii) The change in potential energy of the brick is:
ΔU = mgh
where h is the height difference between the initial and final positions of the brick. We can find h using trigonometry:
h = dsinθ
h = (2.60 m)sin(29.0o) = 1.24 m
Substituting the values into the equation, we get:
ΔU = (4.0 kg)(9.81 m/s^2)(1.24 m)
ΔU = 48.6 J
Therefore, the change in kinetic energy of the brick is 60.1 J and the change in potential energy is 48.6 J.
The figure shows a 4.0-kg brick moving up an inclined plane at an initial speed of vi = 7.50 m/s. The brick comes to rest after traveling a distance d = 2.60 m along a plane that is inclined at an angle of θ = 29.0o to the horizontal. Determine (i) the change in kinetic energy of the brick and (ii) the change in potential energy of the brick.
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