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Chapter 5: Problem 33

W A child and a sled with a combined mass of \(50.0 \mathrm{~kg}\) slide down a frictionless slope. If the sled starts from rest and has a speed of \(3.00 \mathrm{~m} / \mathrm{s}\) at the bottom, what is the height of the hill?

Short Answer

Expert verified
The height of the hill is 0.46 m.

Step by step solution

01

Identify the Given Parameters

The mass (m) of the child and the sled combined is 50.0 kg, the final speed (v) at the bottom of the hill is 3.00 m/s, and we need to find the height (h) of the hill. The acceleration due to gravity (g) is approximately 9.81 m/s\(^2\), which is a known constant.
02

Apply the Principle of Conservation of Energy

The principle of conservation of energy states that the total energy in an isolated system remains constant. Here, the potential energy at the top of the hill is transformed completely into kinetic energy at the bottom, since there are no other types of energy involved and the slope is frictionless. We can express this principle with the equation mgh = \(\frac{1}{2}\)mv\(^2\).
03

Solve the Equation for the Unknown

We are asked to find the height of the hill, so we rearrange the equation to solve for h: h = \(\frac{2*(m*v^2)}{(m*g)}\). We can simplify this equation, as the mass (m) of the sled and child cancels out on both sides: h = \(\frac{2*v^2}{g}\).
04

Substitute the Given Values into the Equation

Substitute v = 3.00 m/s and g = 9.81 m/s\(^2\) into the equation: h = \(\frac{2*(3.00 m/s)^2}{9.81 m/s^2}\), and we calculate the height (h).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Potential Energy
When we talk about potential energy in the context of a child and sled sliding down a slope, we're referring to the energy stored due to their position at the top. Imagine holding a ball up high before dropping it; the higher you hold it, the more potential energy it has. Similarly, the child and sled at the top of the hill possess gravitational potential energy, which can be calculated with the formula PE = mgh, where m is mass, g is the acceleration due to gravity, and h is height.

Kinetic Energy
Once the child and sled begin their descent, their potential energy is converted into kinetic energy, which is the energy of motion. The faster they move, the greater their kinetic energy. For any object with mass m and speed v, kinetic energy (KE) is given by the formula KE = 1/2 mv^2. So as the sled picks up speed sliding down the hill, its kinetic energy increases, reaching its maximum when potential energy has fully transformed into kinetic energy at the bottom of the hill.

Frictionless Slope Physics
The problem specifies a frictionless slope, a simplification often used in physics to focus on the essentials of energy conversion. In reality, frictional forces always exist, but in our model, we assume that they don't affect the sled's motion. This means all the potential energy converts into kinetic energy without any loss, allowing us to use the conservation of energy principle to relate the height of the hill to the speed of the sled at the bottom with precision.

Gravitational Acceleration
The role of gravitational acceleration, denoted by g, is crucial as it determines the rate at which the sled accelerates as it slides down the hill. On Earth, g is approximately 9.81 m/s^2. This constant acceleration due to gravity affects the speed of the sled and consequently its kinetic energy. Since our scenario is idealized with no air resistance or friction, the sled's acceleration remains constant, allowing for the straightforward application of energy conservation to calculate the hill's height.

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Most popular questions from this chapter

In the dangerous "sport" of bungee jumping, a daring student jumps from a hot- air balloon with a specially designed elastic cord attached to his waist. The unstretched length of the cord is \(25.0 \mathrm{~m}\), the student weighs \(700 \mathrm{~N}\), and the balloon is \(36.0 \mathrm{~m}\) above the surface of a river below. Calculate the required force constant of the cord if the student is to stop safely \(4.00 \mathrm{~m}\) above the river.

M A 2 100-kg pile driver is used to drive a steel I-beam into the ground. The pile driver falls \(5.00 \mathrm{~m}\) before coming into contact with the top of the beam, and it drives the beam \(12.0 \mathrm{~cm}\) farther into the ground as it comes to rest. Using energy considerations, calculate the average force the beam exerts on the pile driver while the pile driver is brought to rest.

An 80.0-kg skydiver jumps out of a balloon at an altitude of \(1000 \mathrm{~m}\) and opens the parachute at an altitude of \(200.0 \mathrm{~m}\). (a) Assuming that the total retarding force on the diver is constant at \(50.0 \mathrm{~N}\) with the parachute closed and constant at \(3600 \mathrm{~N}\) with the parachute open, what is the speed of the diver when he lands on the ground? (b) Do you think the skydiver will get hurt? Explain. (c) At what height should the parachute be opened so that the final speed of the skydiver when he hits the ground is \(5.00 \mathrm{~m} / \mathrm{s}\) ? (d) How realistic is the assumption that the total retarding force is constant? Explain. \(5.6\) Power

GP A \(60.0-\mathrm{kg}\) athlete leaps straight up into the air from a trampoline with an initial speed of \(9.0 \mathrm{~m} / \mathrm{s}\). The goal of this problem is to find the maximum height she attains and her speed at half maximum height. (a) What are the interacting objects and how do they interact? (b) Select the height at which the athlete's speed is \(9.0 \mathrm{~m} / \mathrm{s}\) as \(y=0\). What is her kinetic energy at this point? What is the gravitational potential energy associated with the athlete? (c) What is her kinetic energy at maximum height? What is the gravitational potential energy associated with the athlete? (d) Write a general equation for energy conservation in this case and solve for the maximum height. Substitute and obtain a numerical answer. (c) Write the general equation for energy conservation and solve for the velocity at half the maximum height. Substitute and obtain a numerical answer.

A weight lifter lifts a \(350-N\) set of weights from ground level to a position over his head, a vertical distance of \(2.00 \mathrm{~m}\). How much work does the weight lifter do, assuming he moves the weights at constant speed?
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