Alisha starts pulling (from rest) a box full of books on a horizontal floor. She has tied the rope to the box, the rope makes an angle of 40.0° relative to the floor. She pulls with a force F of magnitude 120N. The total mass of the box is 50.0kg. She pulls the box in a straight line over a distance of 6.00m. There is 70.0N friction force between the floor and the box. Calculate the final speed of the box.

To calculate the final speed of the box, we can use Newton's second law and the work-energy principle. Here are the steps to find the solution:

Step 1: Resolve the forces

First, let's resolve the force F into its horizontal and vertical components. The horizontal component of the force (F_h) will be F * cos(40°), and the vertical component (F_v) will be F * sin(40°). Since we are interested in the horizontal motion of the box, we will only consider the horizontal forces.

Step 2: Calculate the net force

Next, we need to calculate the net force acting on the box. The net force is the vector sum of all forces acting in the horizontal direction. In this case, the only horizontal force is the force F_h, since there is no horizontal component of the friction force. Thus, the net force (F_net) is equal to F_h.

F_net = F_h = F * cos(40°)

Step 3: Calculate the acceleration

We can use Newton's second law, which states that the net force acting on an object is equal to its mass multiplied by its acceleration (F_net = m * a), to find the acceleration (a) of the box. Rearranging the equation, we have:

a = F_net / m

Step 4: Calculate the work done

To calculate the work done on the box, we can use the work-energy principle, which states that the work done on an object is equal to the change in its kinetic energy. In this case, the work done (W) is equal to the force applied (F_net) multiplied by the distance (d) the force is applied over:

W = F_net * d

Step 5: Calculate the change in kinetic energy

Since the work done is equal to the change in kinetic energy, we can write:

W = ΔKE

Step 6: Calculate final speed

The change in kinetic energy can be expressed in terms of the initial kinetic energy (KE_initial) and the final kinetic energy (KE_final):

ΔKE = KE_final - KE_initial

Since the box starts from rest, the initial kinetic energy is zero. Therefore, we can simplify the equation to:

ΔKE = KE_final

Combining this with the expression for work done, we have:

F_net * d = KE_final

Finally, we can calculate the final speed (v_final) using the equation:

KE_final = (1/2) * m * v_final^2

By rearranging the equation, we can solve for v_final:

v_final = √(2 * KE_final / m)

Now, let's plug in the given values and calculate the final speed of the box.