Use the work-energy theorem to solve each of these problems....

Use the work-energy theorem to solve each of these problems. You can use Newton's laws to check your answers. Neglect air resistance in all cases. (a) A branch falls from the top of a 95.0-m-tall redwood tree, starting from rest. How fast is it moving when it reaches the ground? (b) A volcano ejects a boulder directly upward 525 $\mathrm{m}$ into the air. How fast was the boulder moving just as it left the volcano? (c) A skier moving at 5.00 $\mathrm{m} / \mathrm{s}$ encounters a long, rough horizontal patch of snow having coefficient of kinetic friction 0.220 with her skis. How far does she travel on this patch before stopping? (d) Suppose the rough patch in part (c) was only 2.90 $\mathrm{m}$ long? How fast would the skier be moving when she reached the end of the patch? (e) At the base of a frictionless icy hill that rises at $25.0^{\circ}$ above the horizontal, a toboggan has a speed of 12.0 $\mathrm{m} / \mathrm{s}$ toward the hill. How high vertically above the base will it go before stopping?

Key Concepts

Friction and Non-Conservative Forces

Friction is a non-conservative force that dissipates mechanical energy, typically converting it into thermal energy. The work done by friction is calculated as the force of friction multiplied by the distance over which it acts, and it must be included in energy analyses to accurately predict motion in situations where objects encounter surfaces with kinetic friction.

Newton's Laws

Newton's laws, particularly the second law which relates force, mass, and acceleration, provide an alternative framework to analyze motion. They serve as a cross-check for problems solved using the work-energy theorem, ensuring that energy principles and force dynamics yield consistent results.

Conservation of Mechanical Energy

The conservation of mechanical energy principle states that the total mechanical energy (the sum of kinetic and potential energy) remains constant in the absence of non-conservative forces like friction. When non-conservative forces are present, their work must be accounted for as they either add to or remove energy from the system.

Work-Energy Theorem

The work-energy theorem states that the net work done by all forces acting on an object is equal to the change in its kinetic energy. This principle allows for the determination of an object’s speed or displacement by equating the work done by forces (including gravitational force, friction, etc.) to the change in kinetic energy without needing to analyze the motion step-by-step.

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field. It is calculated by the product of the object's mass, gravitational acceleration, and height above a reference level. Changes in gravitational potential energy play a key role in problems where objects move vertically, such as falling or being projected upward.

Kinetic Energy

Kinetic energy is the energy an object has due to its motion, typically expressed as one-half of the mass times the square of its velocity. It is a central concept in applying the work-energy theorem, as the work done on an object results in changes in its kinetic energy.

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