Answer: 288.8 m
Explanation:
We have the following data:
[tex]t=15.2 s[/tex] is the time it takes to the child to reach the bottom of the slope
[tex]V_{o}=0[/tex] is the initial velocity (the child started from rest)
[tex]\theta=15\°[/tex] is the angle of the slope
[tex]d[/tex] is the length of the slope
Now, the Force exerted on the sled along the ramp is:
[tex]F=ma[/tex] (1)
Where [tex]m[/tex] is the mass of the sled and [tex]a[/tex] its acceleration
In addition, if we draw a free body diagram of this sled, the force along the ramp will be:
[tex]F=mg sin \theta[/tex] (2)
Where [tex]g=9.8 m/s^{2}[/tex] is the acceleration due gravity
Then:
[tex]ma=mg sin \theta[/tex] (3)
Finding [tex]a[/tex]:
[tex]a=g sin \theta[/tex] (4)
[tex]a=9.8 m/s^{2} sin(15\°)[/tex] (5)
[tex]a=2.5 m/s^{2}[/tex] (6)
Now, we will use the following kinematic equations to find [tex]d[/tex]:
[tex]V=V_{o}+at[/tex] (7)
[tex]V^{2}=V_{o}^{2}+2ad[/tex] (8)
Where [tex]V[/tex] is the final velocity
Finding [tex]V[/tex] from (7):
[tex]V=at=(2.5 m/s^{2})(15.2 s)[/tex] (9)
[tex]V=38 m/s[/tex] (10)
Substituting (10) in (8):
[tex](38 m/s)^{2}=2(2.5 m/s^{2})d[/tex] (11)
Finding [tex]d[/tex]:
[tex]d=288.8 m[/tex]
