Question

A 3.7 kg box is pushed so that it is moving at a speed of 10.5 m/s when it begins up an incline of 25 degrees. If the coefficient of friction is 0.22, how far up the ramp will the box travel?

Answer 1

To determine how far the box will travel up the ramp, we need to analyze the forces acting on the box and calculate the work done. We'll assume there is no air resistance.

Let's break down the forces acting on the box:

1. Weight (mg): The weight of the box is given by the formula: weight = mass × gravitational acceleration. The gravitational acceleration is approximately 9.8 m/s². Therefore, the weight of the box is: weight = 3.7 kg × 9.8 m/s² = 36.26 N.

2. Normal force (N): The normal force acts perpendicular to the surface of the incline. It is equal in magnitude but opposite in direction to the component of the weight perpendicular to the incline. Therefore, N = weight × cos(θ), where θ is the angle of the incline (25 degrees in this case).

3. Force of friction (f): The force of friction opposes the motion of the box up the incline. Its magnitude is given by the formula: f = coefficient of friction × N.

4. Net force (F_net): The net force acting on the box is the component of the weight parallel to the incline (mg × sin(θ)) minus the force of friction (f).

Now, let's calculate the net force:

Net force (F_net) = (weight × sin(θ)) - f

Substituting the known values:

F_net = (36.26 N × sin(25°)) - (0.22 × N)

To find the acceleration of the box, we use Newton's second law: F_net = mass × acceleration. Rearranging the equation:

acceleration = F_net / mass

Substituting the values:

acceleration = (36.26 N × sin(25°)) - (0.22 × N) / 3.7 kg

Now we can calculate the acceleration.

Next, we need to determine the time it takes for the box to come to rest. When the box comes to rest, its final velocity is zero. We can use the equation of motion: final velocity = initial velocity + (acceleration × time). In this case, the final velocity is 0 m/s, the initial velocity is 10.5 m/s, and we have the acceleration calculated previously. Rearranging the equation:

time = (final velocity - initial velocity) / acceleration

Finally, we can calculate the distance traveled up the ramp using the equation of motion: distance = initial velocity × time + (0.5 × acceleration × time²).

By substituting the known values into these equations, we can calculate the distance up the ramp traveled by the box.