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 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.
To determine the change in kinetic energy of the brick, we need to calculate the final velocity (vf) of the brick when it comes to a stop.
We can use the equation for final velocity (vf) on an inclined plane based on the initial velocity (vi), distance traveled (d), and the angle of inclination (θ) to the horizontal.
The equation is given as:
vf^2 = vi^2 + 2ad
Where:
vi = initial velocity = 7.50 m/s
d = distance traveled = 2.60 m
θ = angle of inclination = 29.0°
Let's calculate the final velocity (vf) first.
Using the given values in the equation:
vf^2 = vi^2 + 2ad
vf^2 = (7.50 m/s)^2 + 2 * (4.0 kg) * (9.8 m/s^2) * (2.60 m * sin(29.0°))
We can simplify the equation using trigonometric functions:
vf^2 = (7.50 m/s)^2 + 2 * (4.0 kg) * (9.8 m/s^2) * (2.60 m * 0.482)
Calculating:
vf^2 = 56.25 m^2/s^2 + 38.176 m^2/s^2
vf^2 = 94.426 m^2/s^2
Taking the square root of both sides, we get:
vf = √94.426 m^2/s^2
vf ≈ 9.72 m/s (rounded to two decimal places)
Now that we have the final velocity, we can calculate the change in kinetic energy (ΔKE) of the brick using the formula:
ΔKE = (1/2) * m * (vf^2 - vi^2)
Where:
m = mass of the brick = 4.0 kg
vi = initial velocity = 7.50 m/s
vf = final velocity ≈ 9.72 m/s
Using the given values in the formula:
ΔKE = (1/2) * (4.0 kg) * (9.72 m/s)^2 - (7.50 m/s)^2
Calculating:
ΔKE = (1/2) * 4.0 kg * 94.4544 m^2/s^2 - 56.25 m^2/s^2
ΔKE ≈ 189.1736 J - 56.25 J
ΔKE ≈ 132.9236 J
Therefore, the change in kinetic energy of the brick is approximately 132.9236 Joules.
To determine the change in potential energy of the brick, we need to calculate the initial potential energy (PEi) and the final potential energy (PEf) of the brick.
The equation for potential energy on an inclined plane is given as:
PE = m * g * h
Where:
m = mass of the brick = 4.0 kg
g = acceleration due to gravity = 9.8 m/s^2
h = vertical height change
The change in vertical height (h) can be calculated using the distance traveled (d) along the inclined plane and the angle of inclination (θ) to the horizontal.
The equation is given as:
h = d * sin(θ)
Using the given values, we can calculate h:
h = 2.60 m * sin(29.0°)
Calculating:
h = 2.60 m * 0.482
h ≈ 1.2552 m (rounded to four decimal places)
The initial potential energy (PEi) of the brick is given by:
PEi = m * g * h
Using the given values:
PEi = 4.0 kg * 9.8 m/s^2 * 1.2552 m
Calculating:
PEi = 49.0192 J
The final potential energy (PEf) of the brick is given by:
PEf = m * g * h
Using the given values:
PEf = 4.0 kg * 9.8 m/s^2 * 0
Calculating:
PEf = 0 J
Therefore, the change in potential energy of the brick is the difference between the initial and final potential energies:
ΔPE = PEf - PEi
Calculating:
ΔPE = 0 J - 49.0192 J
ΔPE ≈ -49.0192 J
Therefore, the change in potential energy of the brick is approximately -49.0192 Joules.