Question

A sled slides down a hill 42 meters high with a slope of 27 o. The total mass of the sled plus contents is 256 kg. It starts from rest. There is no friction. Use the Work Energy Theorem to find the velocity of the sled at the bottom.

Answer 1

Answer:

v = 28.7 m/s

Explanation:

• According to the Work Energy Theorem, the work done by external forces on the system, is equal to the change in the kinetic energy of the system.
• In absence of friction, the only force that does work , producing a displacement in the direction of the force, is the component of gravity parallel to the slide:

       $Fg_{p } = m*g* sin \theta (1)$

• The displacement d,along the slide, can be found from the definition of the sine of an angle, as follows:

        $sin \theta =\frac{h}{d}$

• Solving for d, we get:

       $d = \frac{h}{sin \theta} (2)$

• Now, the work done by Fgp, is just the product of the force times the displacement, as follows:

        $W = Fgp * d$

• From (1) and (2) we can find W as follows:

       $W =Fg_{p }* d = m*g* sin \theta * \frac{h}{sin \theta} = m*g*h$

• This expression must be equal to ΔK, as follows:

        $\Delta K = \frac{1}{2} * m *v^{2} = m*g*h$

• Simplifying common terms, we can solve for v (the velocity of the sled at the bottom of the slide), as follows:

        $v =\sqrt{2*g*h} = \sqrt{2*9.8 m/s2*42 m} = 28.7 m/s$

• The velocity of the sled at the bottom of the slide is 28.7 m/s, taking as positive the direction down the slide.