You and your friend peter are putting new shingles on a roof pitched at 25 degrees. you are sitting on the very top of the roof when peter who is at the edge of the roof directly below you, 5.0m away, asks you for a box of nails. rather than carry the 2.5kg box of nails down to peter, you decide to give the box a push and have it slide down to him. if the coefficient of the kinetic friction between the box and the roof is 0.55, with what speed should you push the box to have it gently come to rest right at the edge of the roof?

Newton's second law and kinematics we can find the launch speed of the box is:

Newton's second law gives a relationship between the force, mass, and acceleration of the body.

       ∑ F = m a

Where the bold letters indicate vectors, m is the mass and the acceleration.

A reference system is a coordinate system with respect to which measurements are made, in this case the x-axis is placed along the plane and the positive direction the direction of movement, the  y-axis is placed perpendicular to the plane.

In the attached we see a free body diagram of the system, this free body diagram is a diagram of the forces without the details of the bodies.

Let's write Newton's second law for each axis.

y-axis

       N-  = 0

       N =  

x-axis

       Wₓ -fr = m a

Let's use trigonometry to find the components of weight.

       cos 25 =  

       sin 25 =    

        = W cos 25

       Wₓ = W sin 25

The friction force is a macroscopic force that is the result of the microscopic interactions between the two surfaces, it is expressed.

        fr = μ N

let's replace.

        mg sin 25 - μ m g cos 25 = m a

        a = g (sin 25 - μ cos 25)

Let's calculate.

       a = 9.8 (sin 25 - 0.55 cos 25)

       a = -0.743 m/ s²

The negative sign indicates that the direction is going up the plane.

Now we can use kinematics.

         v² = v₀² - 2a x

They indicate that the box reaches the edge, that is, its final speed is zero.

         0 = v₀² - 2 ax

         v₀ =  

Let's calculate

         v₀ =

          v₀ = 2,726 m / s

In conclusion using Newton's second law and kinematics we can find the launch speed of the box is:

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2.72 m/s

Explanation:

Given

Inclination of shingles

Length of inclined Plane

coefficient of kinetic friction

Let u be the initial velocity

i.e. deceleration