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(solution in the pdf version of the book) A firework shoots up into the air, and just before it explodes it has a certain momentum and kinetic energy. What can you say about the momenta and kinetic energies of the pieces immediately after the explosion? [Based on a problem from PSSC Physics.]

Short Answer

Expert verified
Momentum is conserved; kinetic energy typically increases after an explosion.

Step by step solution

01

Understanding Conservation of Momentum

In physics, the law of conservation of momentum states that the total momentum of a closed system is constant if no external forces are acting on it. Thus, the momentum of the firework just before it explodes is the same as the total momentum of all its pieces immediately after the explosion. Mathematically, this can be expressed as: \( \vec{p}_{\text{initial}} = \sum \vec{p}_{\text{pieces}} \).
02

Considering Conservation of Kinetic Energy

Unlike momentum, kinetic energy is not always conserved in explosions. An explosion is typically an inelastic process that transforms potential energy stored in the firework into kinetic energy of the pieces. Therefore, the total kinetic energy of the pieces after the explosion will be greater than the kinetic energy of the firework just before it exploded. This means \( KE_{\text{initial}} \leq \sum KE_{\text{pieces}} \), although the exact distribution depends on the explosion specifics.
03

Summarizing Effects of Explosion on Motion and Energy

Immediately after the explosion, the momentum is distributed among the various pieces, maintaining the total initial momentum, but their individual velocities and masses determine each piece's momentum. Meanwhile, the conversion of potential energy into kinetic energy means that the sum of the kinetic energies of the pieces will be higher than the initial kinetic energy. Thus, both conservation laws provide insight: momentum is conserved while kinetic energy increases in total.

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Key Concepts

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

Conservation of Kinetic Energy
When discussing conservation of kinetic energy, it's vital to understand that it relates to the energy an object possesses due to its motion. In an ideal scenario, like an elastic collision, kinetic energy remains constant. However, in explosions, things typically work differently. These events often involve inelastic processes where other forms of energy, like chemical or potential energy, convert into kinetic energy.

In the case of a fireworks explosion, before it bursts, what happens is that much of the stored potential energy transforms into the kinetic energy of the pieces. This causes the total kinetic energy after the explosion to be greater than the kinetic energy before it. While momentum is conserved, kinetic energy isnt' always static. Different fragments of the firework may move at various speeds, leading to energy distribution changes.
  • Explosions are inelastic, meaning kinetic energy isn't conserved.
  • Potential energy transforms into kinetic energy.
  • Total kinetic energy after the explosion often surpasses the initial kinetic energy.
Inelastic Collision
An inelastic collision is an event in which two objects collide and kinetic energy is not conserved. As mentioned, explosions, including those in fireworks, are common examples where inelastic collisions occur. During these processes, some of the kinetic energy is converted into other energy forms, such as heat or light, or used to do work like breaking apart the firework.

The crux of inelastic collisions is that while kinetic energy changes, the total momentum is conserved. This means that even though the total movement or speed within the system remains constant, the energy available for motion transforms or gets redistributed.
  • Kinetic energy is not conserved in inelastic collisions.
  • Some energy changes forms (e.g., to heat or sound).
  • Momentum remains conserved, providing a useful analysis pathway.
Momentum Distribution
Momentum distribution involves how momentum spreads out among objects during and after an event like an explosion. Even if pieces fly apart in various directions, the total might before and after the explosion stays the same. The principle of momentum conservation states that the vector sum of all individual pieces' momentum equals the system's initial momentum.

In a firework scenario, each piece carries a portion of the total previous momentum, with the amounts varying based on their velocity and mass. Despite varied directions, everything aligns so that the combined momentum vectors sum to the initial momentum.
  • Movement is distributed among the pieces post-explosion.
  • Total system momentum before equals total among all pieces after.
  • Each piece's momentum depends on its speed and mass.
Explosions in Physics
Explosions are fascinating physical events that illustrate many fundamental physics principles. An explosion happens when stored energy rapidly converts into kinetic energy, causing a sudden and often dramatic increase in volume or spread of objects.

Understanding how explosions work involves studying how energy and momentum are redistributed. While the firework initially might store chemical energy, the ignition transforms it into motion. This rapid change influences how pieces separate and spread. Despite vast kinetic energy differences, momentum conservation plays a key role in determining the motions and interactions of the resulting fragments.
  • Explosions transform stored energy into kinetic energy.
  • Rapid energy conversion causes objects to spread dramatically.
  • Momentum conservation helps predict individual fragment pathways.

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Most popular questions from this chapter

At low speeds, every car's acceleration is limited by traction, not by the engine's power. Suppose that at low speeds, a certain car is normally capable of an acceleration of \(3 \mathrm{~m} / \mathrm{s}^{2}\). If it is towing a trailer with half as much mass as the car itself, what acceleration can it achieve? [Based on a problem from PSSC Physics.]

A gun is aimed horizontally to the west. The gun is fired, and the bullet leaves the muzzle at \(t=0\). The bullet's position vector as a function of time is \(\mathbf{r}=b \hat{\mathbf{x}}+c t \hat{\mathbf{y}}+d t^{2} \hat{\mathbf{z}}\), where \(b, c\), and \(d\) are positive constants. (a) What units would \(b, c\), and \(d\) need to have for the equation to make sense? (b) Find the bullet's velocity and acceleration as functions of time. (c) Give physical interpretations of \(b, c, d, \hat{\mathbf{x}}, \hat{\mathbf{y}}\), and \(\hat{\mathbf{z}}\).

A plane is flown in a loop-the-loop of radius \(1.00 \mathrm{~km}\). The plane starts out flying upside-down, straight and level, then begins curving up along the circular loop, and is right-side up when it reaches the top. (The plane may slow down somewhat on the way up.) How fast must the plane be going at the top if the pilot is to experience no force from the seat or the seatbelt while at the top of the loop? (answer check available at lightandmatter.com)

A ball of mass \(2 m\) collides head-on with an initially stationary ball of mass \(m\). No kinetic energy is transformed into heat or sound. In what direction is the mass- \(2 m\) ball moving after the collision, and how fast is it going compared to its original velocity? \hwans\\{hwans:twotoonecollision\\}

In a well known stunt from circuses and carnivals, a motorcyclist rides around inside a big bowl, gradually speeding up and rising higher. Eventually the cyclist can get up to where the walls of the bowl are vertical. Let's estimate the conditions under which a running human could do the same thing. (a) If the runner can run at speed \(v\), and her shoes have a coefficient of static friction \(\mu_{s}\), what is the maximum radius of the circle?(answer check available at lightandmatter.com) (b) Show that the units of your answer make sense. (c) Check that its dependence on the variables makes sense. (d) Evaluate your result numerically for \(v=10 \mathrm{~m} / \mathrm{s}\) (the speed of an olympic sprinter) and \(\mu_{s}=5\). (This is roughly the highest coefficient of static friction ever achieved for surfaces that are not sticky. The surface has an array of microscopic fibers like a hair brush, and is inspired by the hairs on the feet of a gecko. These assumptions are not necessarily realistic, since the person would have to run at an angle, which would be physically awkward.)(answer check available at lightandmatter.com)

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