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
First, we need to find the final speed of the brick when it comes to rest. We can use the kinematic equation:
vf^2 = vi^2 + 2ad
where vf is the final speed (which is 0 since the brick comes to rest), vi is the initial speed (7.50 m/s), a is the acceleration of the brick (which we can find using the angle of the inclined plane), and d is the distance traveled (2.60 m).
The acceleration of the brick is given by:
a = g*sinθ
where g is the acceleration due to gravity (9.81 m/s^2) and θ is the angle of the inclined plane (29.0o). Substituting in the values, we get:
a = 9.81*sin(29.0o) = 4.74 m/s^2
Now we can use the kinematic equation:
0 = 7.50^2 + 2*(-4.74)*d
Solving for d, we get:
d = 2.24 m
This is the distance that the brick would travel if it came to rest due to friction. Since the actual distance traveled is 2.60 m, we can assume that there is also a component of the acceleration due to gravity acting on the brick, pulling it downwards along the inclined plane.
The change in kinetic energy of the brick can be found using the equation:
ΔK = (1/2)*m*(vf^2 - vi^2)
where m is the mass of the brick (4.0 kg). Substituting in the values, we get:
ΔK = (1/2)*4.0*(0 - 7.50^2) = -140.6 J
The negative sign indicates that the kinetic energy of the brick has decreased.
The change in potential energy of the brick can be found using the equation:
ΔU = m*g*(h2 - h1)
where g is the acceleration due to gravity (9.81 m/s^2), and h2 and h1 are the final and initial heights of the brick, respectively. We can find h2 and h1 by using the trigonometric relationship between the inclined plane and the horizontal.
h2 = d*sinθ = 2.60*sin(29.0o) = 1.35 m
h1 = 0
Substituting in the values, we get:
ΔU = 4.0*9.81*(1.35 - 0) = 52.8 J
The positive sign indicates that the potential energy of the brick has increased.