https://file.aiquickdraw.com/mathbot/file/1729722852827vpq39upl.jpg

We are asked to find the block's speed when it reaches the bottom of the incline. Let's break this down step-by-step.

### Given:
- Mass of the block, \( m = 2.0 \, \text{kg} \)
- Initial speed, \( v_i = 1.0 \, \text{m/s} \)
- Height above the bottom, \( h = 3.0 \, \text{m} \)
- Length of the incline, \( d = 5.0 \, \text{m} \)
- Coefficient of kinetic friction, \( \mu_k = 0.75 \)
- Acceleration due to gravity, \( g = 9.8 \, \text{m/s}^2 \)

### Steps:

1. **Calculate the gravitational force component along the incline:**
   The block is sliding down the incline due to gravity, which has a component along the incline.
   
   The angle of the incline \( \theta \) can be found using:
   \[
   \sin \theta = \frac{h}{d} = \frac{3.0}{5.0} = 0.6
   \quad \Rightarrow \quad \theta = \sin^{-1}(0.6)
   \]
   Now, the component of gravitational force along the incline:
   \[
   F_{\text{gravity}} = mg \sin \theta
   \]

2. **Frictional force:**
   The frictional force opposing the motion is:
   \[
   F_{\text{friction}} = \mu_k mg \cos \theta
   \]
   Here, \( \cos \theta = \sqrt{1 - \sin^2 \theta} = \sqrt{1 - 0.6^2} = 0.8 \).

3. **Net force and acceleration:**
   The net force along the incline is the difference between the gravitational force and the frictional force:
   \[
   F_{\text{net}} = F_{\text{gravity}} - F_{\text{friction}} = mg \sin \theta - \mu_k mg \cos \theta
   \]
   The net acceleration \( a \) is:
   \[
   a = \frac{F_{\text{net}}}{m} = g (\sin \theta - \mu_k \cos \theta)
   \]

4. **Using kinematic equations:**
   To find the final velocity \( v_f \), we use the kinematic equation:
   \[
   v_f^2 = v_i^2 + 2ad
   \]
   Solving for \( v_f \):
   \[
   v_f = \sqrt{v_i^2 + 2ad}
   \]

Now, let’s calculate the individual components.

### 1. Calculate the angle \( \theta \):
\[
\theta = \sin^{-1}(0.6) = 36.87^\circ
\]

### 2. Gravitational and frictional forces:
\[
F_{\text{gravity}} = 2.0 \times 9.8 \times \sin(36.87^\circ) = 11.76 \, \text{N}
\]
\[
F_{\text{friction}} = 0.75 \times 2.0 \times 9.8 \times \cos(36.87^\circ) = 11.76 \, \text{N}
\]

### 3. Net force and acceleration:
\[
F_{\text{net}} = 11.76 - 11.76 = 0 \, \text{N}
\]
\[
a = g (\sin \theta - \mu_k \cos \theta) = 

A 2.0 kg block slides down a flat incline. At one instant, the block is 3.0 m above the bottom of the incline and has a speed of 1.0 m/s down the incline. If the incline is 5.0 m long, and the coefficient of kinetic friction between the block and the incline is μk = 0.75, what is the block’s speed when it reaches the bottom of the incline?

Learn how to find the final speed of a 2.0 kg block sliding down a 5.0 m incline with a friction coefficient of μk = 0.75. The problem involves kinematics, friction, and energy principles. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation