To determine the final velocity of the sled just before it reaches the bottom of the hill, we will break down the problem into several steps:
1. Given Information:
- The inclined angle of the hill, [tex]\(\theta = 38.0^{\circ}\)[/tex]
- The time taken to reach the bottom of the hill, [tex]\( t = 4.24 \, \text{s} \)[/tex]
- The acceleration due to gravity, [tex]\( g = 9.81 \, \text{m/s}^2 \)[/tex]
2. Convert the angle from degrees to radians:
To use trigonometric functions, we need the angle in radians.
[tex]\[
\theta \text{ (in radians)} = 0.6632251157578453
\][/tex]
3. Calculate the acceleration along the incline:
The effective acceleration [tex]\( a \)[/tex] down the incline can be calculated using the component of gravitational acceleration along the slope, which is given by [tex]\( g \sin(\theta) \)[/tex].
[tex]\[
a = 6.039639072944708 \, \text{m/s}^2
\][/tex]
4. Determine the initial velocity:
Since the sled starts from rest, the initial velocity [tex]\( u \)[/tex] is:
[tex]\[
u = 0 \, \text{m/s}
\][/tex]
5. Use the kinematic equation to find the final velocity:
We use the equation [tex]\( v = u + at \)[/tex] where:
- [tex]\( v \)[/tex] is the final velocity.
- [tex]\( u \)[/tex] is the initial velocity, which is [tex]\( 0 \, \text{m/s} \)[/tex].
- [tex]\( a \)[/tex] is the acceleration along the incline.
- [tex]\( t \)[/tex] is the time taken.
Plugging in the known values:
[tex]\[
v = 0 + 6.039639072944708 \, \text{m/s}^2 \times 4.24 \, \text{s}
\][/tex]
[tex]\[
v = 25.608069669285566 \, \text{m/s}
\][/tex]
So, the final velocity of the sled just before it reaches the bottom of the hill is:
[tex]\[
v_f = 25.608069669285566 \, \text{m/s}
\][/tex]
