To determine the acceleration of a car coasting up a frictionless hill inclined at [tex]\(14.0^{\circ}\)[/tex], we can break down the problem step-by-step:
1. Understand the Components of Gravity:
- Gravity acts vertically downward with an acceleration due to gravity, [tex]\(g\)[/tex], which is [tex]\(9.81 \, \text{m/s}^2\)[/tex].
2. Inclination and Its Effect:
- The inclined plane is at an angle of [tex]\(14.0^{\circ}\)[/tex] relative to the horizontal.
3. Decompose Gravity Along the Slope:
- The component of gravitational acceleration along the inclined plane can be found using trigonometry. Specifically, acceleration along the slope, [tex]\(a\)[/tex], is determined by:
[tex]\[
a = g \sin(\theta)
\][/tex]
where [tex]\( \theta = 14.0^\circ \)[/tex] and [tex]\( g = 9.81 \, \text{m/s}^2 \)[/tex].
4. Convert the Angle:
- The sine function will use the angle in radians for computational purposes.
[tex]\[
14.0^{\circ} \approx 0.2443461 \, \text{radians}
\][/tex]
5. Calculate the Gravitational Component Along the Slope:
- Substitute the values into the formula:
[tex]\[
a = 9.81 \, \text{m/s}^2 \times \sin(0.2443461)
\][/tex]
6. Result:
- After performing these calculations, we get the component of the acceleration along the slope as:
[tex]\[
a \approx 2.373 \, \text{m/s}^2
\][/tex]
Thus, the acceleration of the car up the frictionless hill inclined at [tex]\( 14.0^{\circ}\)[/tex] is approximately:
[tex]\[
a \approx 2.373 \, \text{m/s}^2
\][/tex]
