Question24. (II) A $68-\mathrm{kg}$ snowboarder has an initial velocity of $5.0 \mathrm{~m} / \mathrm{s}$ at the top of a $28^{\circ}$ incline (Fig. 5-37). After sliding down the $110-\mathrm{m}$ -long incline (assume a coefficient of kinetic friction $\mu_{\mathrm{k}}=0.18$ ), the snowboarder has attained a velocity $v$ . The snowboarder then slides along a flat surface (on which $\left.\mu_{\mathrm{k}}=0.15\right)$ 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. Then use these accelerations to determine $x$ . Ignore air resistance.

Studdy Solution

STEP 1

1. The snowboarder has a mass of $68 \, \text{kg}$ .
2. The initial velocity at the top of the incline is $5.0 \, \text{m/s}$ .
3. The incline is at an angle of $28^\circ$ .
4. The length of the incline is $110 \, \text{m}$ .
5. The coefficient of kinetic friction on the incline is $\mu_k = 0.18$ .
6. The coefficient of kinetic friction on the flat surface is $\mu_k = 0.15$ .
7. Air resistance is ignored.
8. We need to find the snowboarder's acceleration on the incline and the flat surface, and determine the distance $x$ on the flat surface.

STEP 2

1. Calculate the snowboarder's acceleration on the incline.
2. Calculate the snowboarder's velocity at the bottom of the incline.
3. Calculate the snowboarder's acceleration on the flat surface.
4. Determine the distance $x$ on the flat surface where the snowboarder comes to rest.

STEP 3

Calculate the snowboarder's acceleration on the incline.
The forces acting on the snowboarder along the incline are gravity and friction. The component of gravitational force along the incline is $mg \sin \theta$ , and the frictional force is $\mu_k mg \cos \theta$ .
Using Newton's second law, the net force $F_{\text{net}}$ along the incline is:
$F_{\text{net}} = mg \sin \theta - \mu_k mg \cos \theta$
The acceleration $a$ on the incline is given by:
$a = \frac{F_{\text{net}}}{m} = g \sin \theta - \mu_k g \cos \theta$
Substitute $g = 9.8 \, \text{m/s}^2$ , $\theta = 28^\circ$ , and $\mu_k = 0.18$ :
$a = 9.8 \sin(28^\circ) - 0.18 \times 9.8 \cos(28^\circ)$
Calculate $a$ .

STEP 4

Calculate the snowboarder's velocity at the bottom of the incline.
Use the kinematic equation:
$v^2 = u^2 + 2a s$
where $u = 5.0 \, \text{m/s}$ is the initial velocity, $a$ is the acceleration calculated in STEP_1, and $s = 110 \, \text{m}$ is the distance along the incline.
Solve for $v$ :
$v = \sqrt{u^2 + 2a s}$
Calculate $v$ .

STEP 5

Calculate the snowboarder's acceleration on the flat surface.
The only force acting on the snowboarder on the flat surface is friction, which is $\mu_k mg$ .
Using Newton's second law, the acceleration $a_{\text{flat}}$ is:
$a_{\text{flat}} = \frac{\mu_k mg}{m} = \mu_k g$
Substitute $\mu_k = 0.15$ and $g = 9.8 \, \text{m/s}^2$ :
$a_{\text{flat}} = 0.15 \times 9.8$
Calculate $a_{\text{flat}}$ .

STEP 6

Determine the distance $x$ on the flat surface where the snowboarder comes to rest.
Use the kinematic equation:
$v^2 = u^2 + 2a_{\text{flat}} x$
where $v = 0$ (final velocity), $u$ is the velocity at the bottom of the incline calculated in STEP_2, and $a_{\text{flat}}$ is the acceleration on the flat surface calculated in STEP_3.
Solve for $x$ :
$0 = u^2 + 2a_{\text{flat}} x$ $x = -\frac{u^2}{2a_{\text{flat}}}$
Calculate $x$ .

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms