A crate of mass 10 kg is pulled up a rough incline with an initial speed of 1.5 m/s. The pulling force is 100 N, parallel to the incline; which makes an angle of 20o with the horizontal. The coefficient of friction is 0.40, and the crate is pulled 5.0 m. Has the mechanical energy of the system increased or decreased? Show your work.

potential energy m g h went up. h is 5 sin 20 so change in PE = 10 *9.81 *1.71 = 168 Joules

what happened to kinetic energy .5 m v^2 ?
force up slope = 100 - mu *m*g* cos 20
force up slope = 100 - .4 *10*9.81* .94 = 100-36.9 = 63.1 Newtons
acceleration up slope = 63.1/10 = 6.31 m/s^2
v = 1.5 + a t
x = 0 + 1.5 t + .5 (6.31) * t^2 = 5 meters
3.16 t^2 + 1.5 t -5 = 0 , solve quadratic, plus root = 1.04 seconds
so v = 1.5 + 6.31(1.04) = 11.6 m/s at top
so BOTH potential energy and Kinetic energy INCREASE

To determine if the mechanical energy of the system has increased or decreased, we need to compare the initial and final mechanical energies.

The initial mechanical energy (E_initial) is given by the sum of the kinetic energy (KE) and potential energy (PE) of the system:
E_initial = KE_initial + PE_initial.

The final mechanical energy (E_final) is given by the sum of the final kinetic energy (KE_final) and potential energy (PE_final) of the system:
E_final = KE_final + PE_final.

If E_final is greater than E_initial, then the mechanical energy of the system has increased. If E_final is less than E_initial, then the mechanical energy of the system has decreased.

Let's calculate the initial and final mechanical energy of the system step-by-step:

Step 1: Calculate the initial kinetic energy (KE_initial):
KE_initial = (1/2) * mass * velocity^2

Given: mass = 10 kg, initial velocity (v_initial) = 1.5 m/s
KE_initial = (1/2) * 10 kg * (1.5 m/s)^2
KE_initial = 11.25 J

Step 2: Calculate the potential energy (PE_initial):
PE_initial = mass * g * height

Given: mass = 10 kg, angle of the incline (θ) = 20 degrees, distance pulled (s) = 5.0 m
Height (h) = s * sin(θ)
PE_initial = 10 kg * 9.8 m/s^2 * (5.0 m * sin(20 degrees))
PE_initial = 47.84 J

E_initial = KE_initial + PE_initial
E_initial = 11.25 J + 47.84 J
E_initial = 59.09 J

Step 3: Calculate the final kinetic energy (KE_final):
Since the crate is pulled against friction, some of the initial kinetic energy is lost in overcoming the friction. Therefore, the final kinetic energy will be less than the initial kinetic energy.

Step 4: Calculate the potential energy (PE_final):
PE_final = mass * g * height

Given: mass = 10 kg, angle of the incline (θ) = 20 degrees, distance pulled (s) = 5.0 m
Height (h) = (s * sin(θ)) * cos(θ) (since the distance pulled is along the incline)
PE_final = 10 kg * 9.8 m/s^2 * (5.0 m * sin(20 degrees) * cos(20 degrees))
PE_final = 43.54 J

E_final = KE_final + PE_final

Since the final kinetic energy is less than the initial kinetic energy and the potential energy does not change significantly given the distance pulled, we can conclude that E_final is less than E_initial.

Therefore, the mechanical energy of the system has decreased.

To determine if the mechanical energy of the system has increased or decreased, we need to analyze the work done by various forces and the change in potential and kinetic energy.

First, let's calculate the work done by the applied force parallel to the incline:

Work done by the applied force = Force * distance * cos(theta)
= 100 N * 5.0 m * cos(20°)

Next, let's calculate the work done by the force of friction:

Work done by friction = -Force_friction * distance
= -μ * m * g * distance
= -0.40 * 10 kg * 9.8 m/s^2 * 5.0 m

Now, let's calculate the change in potential energy:

Change in potential energy = m * g * change in height
= 10 kg * 9.8 m/s^2 * change in height

Finally, let's calculate the change in kinetic energy:

Change in kinetic energy = Final kinetic energy - Initial kinetic energy
= 0.5 * m * (final velocity)^2 - 0.5 * m * (initial velocity)^2
= 0.5 * 10 kg * ((0 m/s)^2 - (1.5 m/s)^2)

If the mechanical energy of the system has increased, the work done by the applied force and the change in potential and kinetic energy will all be positive. If it has decreased, one or more of these values will be negative.

Now, let's evaluate each component:

Work done by the applied force = 100 N * 5.0 m * cos(20°) = 921.12 J

Work done by friction = -0.40 * 10 kg * 9.8 m/s^2 * 5.0 m = -196 J

Change in potential energy = 10 kg * 9.8 m/s^2 * change in height

Change in kinetic energy = 0.5 * 10 kg * ((0 m/s)^2 - (1.5 m/s)^2) = -11.25 J

Since the work done by friction is negative and there is a negative change in kinetic energy, it indicates that mechanical energy has decreased.

Therefore, in this case, the mechanical energy of the system has decreased.