A 75-kgkg snowboarder has an initial velocity of 5.0 m/sm/s at the top of a 28 ∘∘ incline. After sliding down the 110-mm long incline (on which the coefficient of kinetic friction is μk�k = 0.18), the snowboarder has attained a velocity v�. The snowboarder then slides along a flat surface (on which μk�k = 0.15) and comes to rest after a distance x�.  Use Newton's second law to find the snowboarder's acceleration while on the incline and while on the flat surface.

To find the snowboarder's acceleration on both the incline and the flat surface, let's apply Newton's second law, \( F = ma \), to each part of the motion separately.

### Step 1: Acceleration on the Incline
Given:
- Mass of the snowboarder, \( m = 75 \, \text{kg} \)
- Incline angle, \( \theta = 28^\circ \)
- Coefficient of kinetic friction on the incline, \( \mu_k = 0.18 \)
- Gravitational acceleration, \( g = 9.81 \, \text{m/s}^2 \)

#### Forces Acting on the Snowboarder on the Incline
1. **Gravitational Force** \( F_g \): The component of gravity acting down the incline is:
   \[
   F_{\text{gravity}} = mg \sin \theta
   \]

2. **Normal Force** \( F_N \): The component of gravity perpendicular to the incline is:
   \[
   F_N = mg \cos \theta
   \]

3. **Frictional Force** \( F_f \): The kinetic friction force is given by:
   \[
   F_f = \mu_k F_N = \mu_k (mg \cos \theta)
   \]

#### Net Force on the Incline
The net force acting on the snowboarder along the incline is:
\[
F_{\text{net}} = F_{\text{gravity}} - F_f = mg \sin \theta - \mu_k (mg \cos \theta)
\]

Using \( F_{\text{net}} = ma \), we can solve for the acceleration \( a \) along the incline:
\[
a = g \sin \theta - \mu_k g \cos \theta
\]

#### Substitute Values
\[
a = (9.81 \, \text{m/s}^2) \sin 28^\circ - (0.18)(9.81 \, \text{m/s}^2) \cos 28^\circ
\]

Let's calculate this value:

1. \(\sin 28^\circ \approx 0.4695\)
2. \(\cos 28^\circ \approx 0.8829\)

Now plug these in:
\[
a \approx (9.81 \times 0.4695) - (0.18 \times 9.81 \times 0.8829)
\]
\[
a \approx 4.6045 - 1.5593
\]
\[
a \approx 3.0452 \, \text{m/s}^2
\]

### Step 2: Acceleration on the Flat Surface
On the flat surface, there is no incline, so the only horizontal force acting on the snowboarder is the kinetic friction force.

Given:
- Coefficient of kinetic friction on the flat surface, \( \mu_k = 0.15 \)
- Normal force on flat ground, \( F_N = mg \)

#### Frictional Force on the Flat Surface
The frictional force is:
\[
F_f = \mu_k F_N = \mu_k mg
\]

#### Net Force on the Flat Surface
Since the frictional force is the only horizontal force, the net force is:
\[
F_{\text{net}} = -F_f = -\mu_k mg
\]

Using \( F_{\text{net}} = ma \), we find:
\[
a = -\mu_k g
\]

#### Substitute Values
\[
a = -(0.15)(9.81)
\]
\[
a \approx -1.4715 \, \text{m/s}^2
\]

### Summary of Results
1. **Acceleration on the incline**: \( a \approx 3.05 \, \text{m/s}^2 \)
2. **Acceleration on the flat surface**: \( a \approx -1.47 \, \text{m/s}^2 \)

Would you like a further breakdown of the calculations or the next steps to solve for the distance and final velocity?

### Related Questions to Expand Understanding
1. How can the final velocity at the bottom of the incline be calculated?
2. How does the angle of the incline affect the snowboarder's acceleration?
3. What would the acceleration be if the coefficient of kinetic friction were zero?
4. How do we calculate the distance required to bring the snowboarder to rest on the flat surface?
5. What role does mass play in determining acceleration in this scenario?

### Tip
Remember, friction always acts opposite to the direction of motion, so its effect will reduce the acceleration on the incline and cause deceleration on the flat surface.

A 75-kg snowboarder has an initial velocity of 5.0 m/s at the top of a 28° incline. After sliding down the 110-m long incline (on which the coefficient of kinetic friction is μk = 0.18), the snowboarder has attained a velocity v. The snowboarder then slides along a flat surface (on which μk = 0.15) and comes to rest after a distance x. Use Newton's second law to find the snowboarder's acceleration while on the incline and while on the flat surface.

Learn to find the acceleration of a 75-kg snowboarder sliding down a 28° incline and a flat surface, with coefficients of kinetic friction of 0.18 and 0.15, respectively. This problem applies Newton's Second Law and explores concepts like gravitational force on inclines, normal force, and kinetic friction. Suitable for students in Grades 10-12.

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