Make a graph of the kinetic energy versus momentum for (a) a particle of nonzero mass, and (b) a particle with zero mass.

Verified step by step guidance

1. Understand the relationship between kinetic energy (K) and momentum (p) for a particle with nonzero mass. The kinetic energy is given by the formula: $K\; =\; \backslash frac\{p^2\}\{2m\}$, where $p$ is the momentum and $m$ is the mass of the particle. This equation shows that kinetic energy is proportional to the square of the momentum for a particle with mass.
2. For a particle with zero mass (e.g., a photon), the relationship between energy and momentum is different. The energy is given by $E\; =\; pc$, where $p$ is the momentum and $c$ is the speed of light. Since kinetic energy is equivalent to the total energy for a massless particle, the graph of kinetic energy versus momentum will be a straight line passing through the origin with slope $c$.
3. To graph the kinetic energy versus momentum for a particle with nonzero mass, plot $K$ on the y-axis and $p$ on the x-axis. The graph will be a parabola opening upwards, as $K\; =\; \backslash frac\{p^2\}\{2m\}$ is a quadratic equation in $p$.
4. To graph the kinetic energy versus momentum for a particle with zero mass, plot $K$ on the y-axis and $p$ on the x-axis. The graph will be a straight line with a slope equal to the speed of light $c$, as $K\; =\; pc$.
5. Compare the two graphs. For a particle with nonzero mass, the kinetic energy increases quadratically with momentum, while for a massless particle, the kinetic energy increases linearly with momentum.

Key Concepts

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion, calculated using the formula KE = 1/2 mv², where m is mass and v is velocity. For particles with nonzero mass, kinetic energy increases with the square of velocity, illustrating how faster-moving objects have significantly more energy.

Momentum

Momentum is a vector quantity defined as the product of an object's mass and its velocity, expressed as p = mv. It is a crucial concept in physics because it is conserved in isolated systems, meaning the total momentum before and after an event remains constant, which is essential for analyzing collisions and interactions.

Massless Particles

Massless particles, such as photons, travel at the speed of light and do not have rest mass. Their momentum is defined differently, as p = E/c, where E is energy and c is the speed of light. This distinction leads to unique relationships between kinetic energy and momentum for massless particles, differing significantly from those of massive particles.