Energy Diagrams, Collisions, and Power: Study Notes for College Physics

Energy Diagrams

Section 10.7: Energy Diagrams

Energy diagrams are graphical representations that show how potential energy varies with position for a physical system. They are essential tools for visualizing and analyzing the motion and equilibrium of objects under conservative forces.

• Energy diagrams plot potential energy (PE) as a function of position.

• For free fall, the energy diagram is a straight line, representing gravitational potential energy:

$U_g = mgy$

• For a spring, the energy diagram is a parabola, representing spring potential energy:

$U_s = \frac{1}{2}kx^2$

• These diagrams help predict motion, equilibrium, and turning points.

Mass Oscillating on a Spring

When a mass oscillates on a spring, its energy diagram is parabolic. The total mechanical energy is conserved (in the absence of non-conservative forces), and the system oscillates between kinetic and potential energy.

• At maximum displacement, all energy is potential ($U_s$ is maximum, $K = 0$).

• At equilibrium (center), all energy is kinetic ($K$ is maximum, $U_s = 0$).

• The motion is periodic, and the energy diagram illustrates the exchange between kinetic and potential energy.

Interpreting an Energy Diagram

Energy diagrams can be used to analyze the motion of an object and determine possible positions, speeds, and turning points.

• At any position, the vertical distance from the axis to the PE curve is the object's potential energy.

• The vertical distance from the PE curve to the total energy line ($E$) is the object's kinetic energy.

• The object cannot exist at positions where the PE curve is above the $E$ line (forbidden regions).

• Where the $E$ line crosses the PE curve are turning points (object reverses direction).

• If the $E$ line crosses the PE curve at two positions, the object oscillates between those positions.

• Speed is maximum where the PE curve is at a minimum (kinetic energy is maximized).

Example: For a particle in a potential well, the lowest point of the well corresponds to the highest speed, as kinetic energy is greatest where potential energy is lowest.

Equilibrium Positions

Equilibrium occurs at positions where the force on the object is zero, which corresponds to local minima or maxima in the potential energy curve.

• Stable equilibrium: Found at local minima of the PE curve. Small displacements result in restoring forces.

• Unstable equilibrium: Found at local maxima of the PE curve. Small displacements result in forces that move the object further from equilibrium.

• At equilibrium, energy is entirely potential, and the object is at rest ($K = 0$).

Energy in Collisions

Section 10.9: Energy in Collisions

Collisions are classified based on whether kinetic energy is conserved. All collisions conserve momentum, but only some conserve mechanical energy.

• Perfectly inelastic collision: Colliding objects stick together and move with a common final velocity. Mechanical energy is not conserved.

• Perfectly elastic collision: Both momentum and mechanical energy are conserved.

• In inelastic collisions, some mechanical energy is converted to thermal energy or other forms.

Example: Railroad Cars Collision

Two train cars collide and couple together. To find the thermal energy created:

• Use conservation of momentum to find final velocity.

• Calculate initial and final kinetic energies:

$K_i = \frac{1}{2}m_1 v_{1i}^2 + \frac{1}{2}m_2 v_{2i}^2$

$K_f = \frac{1}{2}(m_1 + m_2)v_f^2$

• The difference is the thermal energy generated:

$\Delta E_{th} = K_i - K_f$

Elastic Collisions

Elastic collisions conserve both momentum and kinetic energy. The final velocities can be determined using the following equations for a head-on collision:

• Momentum conservation:

$m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}$

• Kinetic energy conservation:

$\frac{1}{2}m_1 v_{1i}^2 + \frac{1}{2}m_2 v_{2i}^2 = \frac{1}{2}m_1 v_{1f}^2 + \frac{1}{2}m_2 v_{2f}^2$

• For a stationary target ($v_{2i} = 0$):

$v_{1f} = \frac{m_1 - m_2}{m_1 + m_2} v_{1i}$ $v_{2f} = \frac{2m_1}{m_1 + m_2} v_{1i}$

Example: In a head-on collision between two identical air hockey pucks, the moving puck stops, and the stationary puck moves off with the initial velocity of the first puck.

Power

Section 10.10: Power

Power quantifies the rate at which energy is transferred or transformed in a system. It is a measure of how quickly work is done or energy is used.

• Definition: Power is the rate of energy transfer:

$P = \frac{\Delta E}{\Delta t}$

• Alternatively, power is the rate at which work is done:

$P = \frac{W}{\Delta t}$

• The SI unit of power is the watt ($1\ \text{W} = 1\ \text{J/s}$).

Output Power of a Force

When a force acts on an object and causes displacement, the rate at which the force does work is the output power:

$P = Fv$

• Where $F$ is the force and $v$ is the velocity of the object in the direction of the force.

Example: Weightlifter's Output Power

A 100 kg weightlifter raises a 190 kg bar to a height of 1.9 m in 1.8 s. The output power is calculated as follows:

• Change in gravitational potential energy:

$\Delta U_g = mg\Delta y = (190\ \text{kg})(9.8\ \text{m/s}^2)(1.9\ \text{m}) = 3540\ \text{J}$

• Power:

$P = \frac{\Delta E}{\Delta t} = \frac{3540\ \text{J}}{1.8\ \text{s}} \approx 2000\ \text{W}$

Summary of General Principles

Basic Energy Model

• Energy can be transformed between different forms within a system.

• Energy can be transferred into or out of a system by:

  • Work: Transfer by mechanical forces.

  • Heat: Nonmechanical transfer from a hotter to a colder object.

Conservation of Energy

When work $W$ is done on a system, the system's total energy changes by the amount of work done:

$\Delta E = \Delta K + \Delta U_g + \Delta U_s + \Delta E_{th} + \Delta E_{chem} + \ldots = W$

For an isolated system ($W = 0$):

$\Delta K + \Delta U_g + \Delta U_s + \Delta E_{th} + \Delta E_{chem} + \ldots = 0$

Solving Energy Problems

1. Strategize: Choose the system and determine initial and final states.

2. Prepare: Draw a before-and-after visual overview.

3. Solve: Use the work-energy equation:

$K_i + U_i + \Delta E_{th} = K_f + U_f + W$

• Use appropriate forms of potential energy.

• If the system is isolated, set $W = 0$.

• If there is no friction or drag, set $\Delta E_{th} = 0$.

• Check that energy is conserved and all values are physically reasonable.

Important Concepts

• Kinetic energy: $K = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$ (translational and rotational)

• Potential energy:

  • Gravitational: $U_g = mgy$

  • Elastic (spring): $U_s = \frac{1}{2}kx^2$

• Thermal energy: Sum of microscopic kinetic and potential energies of molecules. Increases with friction or drag:

$\Delta E_{th} = f_k \Delta x$ (friction) $\Delta E_{th} = D \Delta x$ (drag)

• Work: $W = Fd \cos \theta$ (only the component of force parallel to displacement does work)

Applications

• Energy diagrams are used to analyze physical systems, such as molecular bonds or oscillating masses.

• In perfectly elastic collisions, both mechanical energy and momentum are conserved.

• Power calculations are essential for understanding the rate of energy use in physical activities and machines.

Table: Toy Car Power Comparison (from QuickCheck 10.22)

Car	Mass (g)	Speed (m/s)	Time (s)
A	100	3	2
B	200	2	3
C	300	2	3
D	600	1	3
E	400	2	4

Purpose: This table compares the power output of different toy cars based on their mass, speed, and acceleration time. Car A has the greatest power output due to its combination of high speed and short acceleration time.