You and your friend Peter are putting new shingles on a roof pitched at 21 ∘ . You're sitting on the very top of the roof when Peter, who is on the edge of the roof, 5.1 m away, asks you for the box of nails. Rather than carry the 2.5 kg 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 kinetic friction between the box and the roof is 0.54, with what speed should you push the box to have it gently come to rest right at the edge of the roof?

To determine the initial velocity with which you should push the box of nails so that it comes to rest gently at the edge of the roof, we can apply the principles of energy and friction.

### Given dataTo determine the initial velocity with which you should push the box of nails so that it comes to rest gently at the edge of the roof, we can apply the principles of energy and friction.

### Given 
- Mass of the box of nails, \( m = 2.5 \, \text{kg} \)
- Distance from you to Peter, \( d = 5.1 \, \text{m} \)
- Roof pitch angle, \( \theta = 21^\circ \)
- Coefficient of kinetic friction, \( \mu_k = 0.54 \)
- Acceleration due to gravity, \( g = 9.81 \, \text{m/s}^2 \)

We will solve the problem by considering the forces acting on the box as it slides down the roof. The key forces here are:
1. The component of the gravitational force pulling the box down the slope.
2. The frictional force opposing the motion.

### Step 1: Net force acting on the box
The gravitational force along the slope is given by the component of the weight in the direction of the slope:
\[
F_{\text{gravity}} = m g \sin(\theta)
\]

The frictional force opposing the motion is:
\[
F_{\text{friction}} = \mu_k m g \cos(\theta)
\]

Thus, the net force \( F_{\text{net}} \) acting on the box is:
\[
F_{\text{net}} = F_{\text{gravity}} - F_{\text{friction}}
= m g \sin(\theta) - \mu_k m g \cos(\theta)
\]

### Step 2: Net acceleration
Since \( F = ma \), the net acceleration \( a \) of the box down the slope is:
\[
a = g (\sin(\theta) - \mu_k \cos(\theta))
\]

### Step 3: Use kinematic equation to find the initial velocity
We want the box to come to rest at the edge of the roof after traveling a distance of 5.1 m. We use the following kinematic equation, where the final velocity \( v_f = 0 \), the initial velocity is \( v_i \), and the distance traveled is \( d \):
\[
v_f^2 = v_i^2 + 2 a d
\]
Since \( v_f = 0 \), we solve for \( v_i \):
\[
0 = v_i^2 + 2 a d
\]
\[
v_i^2 = -2 a d
\]
\[
v_i = \sqrt{-2 a d}
\]

Now let's calculate \( a \) and \( v_i \).The initial speed with which you should push the box of nails so that it gently comes to rest at the edge of the roof is approximately \( 3.82 \, \text{m/s} \).

Would you like more details on how the forces affect the motion or have any questions?

Here are five related questions to explore further:
1. What would happen if the angle of the roof were steeper?
2. How does the mass of the box affect the required initial speed?
3. What if the coefficient of kinetic friction were lower, such as on a metal roof?
4. How can we calculate the time it takes for the box to reach the edge?
5. How would air resistance alter the calculation?

**Tip:** Always double-check units and conversions, especially with angles when working with trigonometric functions.

You and your friend Peter are putting new shingles on a roof pitched at 21°. You're sitting on the very top of the roof when Peter, who is on the edge of the roof, 5.1 m away, asks you for the box of nails. Rather than carry the 2.5 kg 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 kinetic friction between the box and the roof is 0.54, with what speed should you push the box to have it gently come to rest right at the edge of the roof?

Learn how to calculate the initial velocity needed to push a 2.5 kg box of nails down a 21° inclined roof with a coefficient of kinetic friction of 0.54, ensuring it comes to rest at the edge. This physics problem involves forces, friction, and kinematic equations, suitable for Grades 11-12 and introductory college physics.

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