A 1,982-kg car starts from rest at the top of a driveway 6.74 m long that is sloped at an angle of 30 degrees with the horizontal. If an average friction force of 2,721 N impedes the motion of the car, find the speed (in m/s) of the car at the bottom of the driveway.Use the approximation that g ≈ 10 m/s2.

Sagot :

In this situation, we cannot apply the law of conservation of energy, as there is friction. For us to solve, let us start by writing the balance equations. We'll have:

[tex]\sum F_x=P*sin(30)-Fat=ma[/tex][tex]\sum F_y=N-P*cos(30)=0[/tex]

In order to find out the acceleration, we can use the first equation:

[tex]a=\frac{P*sin(30)-Fat}{m}=\frac{1982*10*sin(30)-2721}{1982}=3.627\frac{m}{s^2}[/tex]

The car will then suffer this acceleration on the sloped plane. With this, we can calculate its speed by the end using the equations for a uniformly accelerated movement:

[tex]S(t)=S_0+v_0t+\frac{at^2}{2}\Rightarrow6.74=\frac{3.627*t^2}{2}\Rightarrow t=1.928s[/tex]

This is the time the car will take to reach the bottom. By replacing this on the equation for the velocity we get:

[tex]v(t)=v_0+at=0+3.627*1.928=7\frac{m}{s}[/tex]

Then, our final answer is 7 m/s

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