Answer:
distance traveled before coming to rest is: 50 meters
Explanation:
As the girl+sled reach the bottom of the hill and start the horizontal travel with a speed of 7 m/s, there are three forces acting on them (please see attached free body diagram):
1) The weight "w" pointing down and known to be 645 N,
2) the normal force "n" of the snow on the sled+girl system which equals and counteracts the weight since there is no vertical displacement any more,
and 3) the force of friction with the snow, which we can calculate from the formula: [tex]F_f=\mu\,*\,n=0.05*645\,N=32.25\,N[/tex]
Therefore, the net resultant force for this system is that of the force of friction: 32.25 N acting to stop the sled. Since this is the net force, we can estimate what (negative) acceleration is being imparted on the system via Newton's second law (F=m*a) if we know the girl+sled mass.
We can determine such knowing that the weight of girl+sled is 645 N:
mass of girl + sled = [tex]\frac{645}{g} \,kg[/tex]
which results in kilograms given that all other quantities are in SI units.
Now we use it to estimate the acceleration that the friction is imparting on the sled+girl system to stop it: [tex]a=32.25 / \frac{645}{g} =\frac{32.25\ *\ g}{645} \,\frac{m}{s^2} = 0.49\,\frac{m}{s^2}[/tex]
Now that we know the acceleration acting (remember that it is negative - acting against the initial travelling velocity), we can use kinematic equations for an accelerated movement:
[tex]v_f^2-v_i^2=2\,a\,(x_f-x_i)[/tex]
We know that the initial velocity is 7 m/s and the final (when the sled stops) is zero, we also have now the negative acceleration exerted by the friction force: "-0.49 [tex]\frac{m}{s^2}[/tex]", and [tex](x_f-x_i)[/tex] is the distance traveled which is what we want to find. So, we solve for this distance:
[tex]v_f^2-v_i^2=2\,a\,(x_f-x_i)\\\frac{v_f^2-v_i^2}{2\,a} = (x_f-x_i)\\\frac{0-7^2}{2(-0.49)} =(x_f-x_i)\\50\,m=(x_f-x_i)[/tex]
And we know that the answer comes in meters since all quantities used in the formula were given in SI units.
