91Ó°ÊÓ

A child of mass \(m\) starts from rest and slides without friction from a height \(h\) along a curved waterslide (Fig. P5.46). She is launched from a height \(h / 5\) into the pool. (a) Is mechanical energy conserved? Why? (b) Give the gravitational potential energy associated with the child and her kinetic energy in terms of \(m g h\) at the following positions: the top of the waterslide, the launching point, and the point where she lands in the pool. (c) Determinc her initial speed \(v_{0}\) at the launch point in terms of \(g\) and \(h\). (d) Determine her maximum airborne height \(y_{\max }\) in terms of \(h, g\), and the horizontal speed at that height, \(v_{0 x}\) (e) Use the \(x\) -component of the answer to part (c) to climinate \(v_{0}\) from the answer to part (d), giving the height \(y_{\text {emax }}\) in terms of \(g, h\), and the launch angle \(\theta\). (f) Would your answers be the same if the waterslide were not frictionless? Explain.

Step by step solution

01

(a) Is mechanical energy conserved? Why?

The mechanical energy of the system (child and slide) is conserved because there are no non-conservative forces doing work on the system, considering that there is no friction and air resistance is neglected.

02

(b) Gravitational potential energy and kinetic energy at different positions

The gravitational potential energy and kinetic energy at the top of the waterslide are \(mgh\) and 0 respectively. At the launching point, the height becomes h/5, thus the gravitational potential energy becomes \(mg(h/5)\). The lost potential energy is converted into kinetic energy, thus the kinetic energy becomes \(mg(4h/5)\). At the point where she lands in the pool, the height becomes 0, thus the gravitational potential energy becomes 0 and kinetic energy is \(mgh\).

03

(c) Initial speed at the launch point

Using the conservation of energy (potential energy at the top is equal to kinetic energy at the launch point): \(mgh=0.5*m*v_{0}^2\). Solving for \(v_{0}\) gives \(v_{0}=\sqrt{2gh}\).

04

(d) Maximum airborne height

When the child reaches maximum height, all of her kinetic energy is converted into potential energy so \(0.5*m*v_{0x}^2=m*g*y_{\text {max }}\). Solving for \(y_{\text {max }}\) gives \(y_{\text {max }}=v_{0x}^2/(2g)\).

05

(e) Height in terms of launch angle

We know that \(v_{0x}=v_{0} \cdot cos(\theta)\). Substituting this into the equation for \(y_{\text {max }}\) and eliminating \(v_{0}\) will give \(y_{\text {max }} = g*h*cos^2(\theta)\).

06

(f) Effect of friction

If there were friction, the mechanical energy would not be conserved because some energy would be initially converted (via work done) into heat energy due to the frictional force.

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Key Concepts

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

Mechanical Energy

In physics, mechanical energy is the sum of potential energy and kinetic energy present in a system. It describes the energy due to the motion and position of an object. If no external forces like friction or air resistance are doing work on the system, then its mechanical energy is conserved.
In the scenario of a child sliding down a waterslide, we see a classic example of conservation of mechanical energy. As the child moves from a higher point to a lower point, her potential energy decreases while her kinetic energy increases. The overall mechanical energy remains constant, as shifts between these forms occur seamlessly.
Understanding this principle helps to grasp why the child's energy transformations do not lead to a net loss of energy. The system's capacity to conserve energy despite the dynamic changes in position and velocity is fundamental in physics.

Gravitational Potential Energy

Gravitational potential energy is the energy stored in an object due to its position above the ground. It is dependent on the mass ( m ), the acceleration due to gravity ( g ), and the height ( h ) above a reference point.
This is calculated using the formula: U = mgh . In the context of the waterslide:

• On top of the slide, the energy is at its maximum, U = mgh .
• While launching, at height h/5 , the potential energy is U = mg(h/5) .
• Upon reaching the pool, the height becomes zero, leading to a potential energy of zero.

Understanding gravitational potential energy gives insight into how height influences energy levels in a system.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion. It is given by the formula KE = 0.5 m v^2 , where m is the mass and v is the velocity of the object.
For the child on the slide:

• At the top, the kinetic energy is zero because her velocity is zero.
• As she descends, potential energy is converted into kinetic energy.
• At the launch point, the kinetic energy is KE = mg(4h/5) , reflecting increased speed.
• When landing in the pool, her kinetic energy becomes KE = mgh , as all potential energy has been converted.

Kinetic energy illustrates how motion dynamics change energy forms and explains how speed changes as the child moves down the slope.

Frictionless Motion

Frictionless motion occurs when an object moves without any frictional force opposing its motion. This simplifies the analysis of motion problems, as mechanical energy is fully conserved.
In this waterslide scenario, since motion is frictionless:

• The child slides without any energy loss to heat, maintaining total mechanical energy.
• The calculations of speed and energy transformations remain straightforward, ensuring the accuracy of conservation principles.
• If friction were present, energy would dissipate, altering kinetic outcomes and requiring additional forces in calculations.

This concept is crucial for understanding why real-world scenarios often introduce complexities, yet in theoretical physics, frictionless motion provides a clear, uncomplicated framework for studying energy transitions.