To determine the acceleration of the sled sliding down a frictionless hill inclined at [tex]\(28.0^{\circ}\)[/tex], let's break down the problem step by step.
1. Identify Known Values:
- Angle of incline ([tex]\(\theta\)[/tex]): [tex]\(28.0^{\circ}\)[/tex]
- Distance slid by the sled ([tex]\(d\)[/tex]): [tex]\(10.0 \)[/tex] meters
- Gravitational acceleration ([tex]\(g\)[/tex]): [tex]\(9.81 \, \text{m/s}^2\)[/tex]
2. Convert the Angle to Radians:
Since trigonometric functions in physics often use radians, it's practical to convert the angle from degrees to radians.
[tex]\[
\theta_{\text{radians}} = \theta \times \left( \frac{\pi}{180} \right)
\][/tex]
Substituting the given angle:
[tex]\[
\theta_{\text{radians}} = 28.0 \times \left( \frac{\pi}{180} \right) \approx 0.4887 \, \text{radians}
\][/tex]
3. Determine the Component of Gravitational Force:
On an incline, the component of the gravitational force causing acceleration down the hill is [tex]\( g \sin(\theta_{\text{radians}}) \)[/tex].
4. Calculate the Acceleration:
We use the formula for the acceleration of the sled along the incline:
[tex]\[
a = g \sin(\theta_{\text{radians}})
\][/tex]
Plugging in the values:
[tex]\[
a = 9.81 \times \sin(0.4887) \approx 4.6055 \, \text{m/s}^2
\][/tex]
Hence, the acceleration of the sled is [tex]\( \boxed{4.6055 \, \text{m/s}^2} \)[/tex].
