Question

In an elementary-particle experiment, a particle of mass \(m\) is fired, with momentum \(m c\), at a target particle of mass \(2 \sqrt{2} m\). The two particles form a single new particle (completely inelastic collision). Find: a) the speed of the projectile before the collision b) the mass of the new particle c) the speed of the new particle after the collision

Short answer

a) c b) mc c) 2√2m d) m(1 + 2√2)

Step-by-step solution

Identify the Given Information and Unknowns

We are given: - Mass of the projectile \(m\) - Momentum of the projectile \(mc\) - Mass of the target particle \(2\sqrt{2}m\) We are to find: a) The speed of the projectile before the collision b) The mass of the new particle c) The speed of the new particle after the collision

Calculate the Speed of Projectile Before the Collision

We are given the momentum of the projectile which is equal to the product of its mass and velocity (\(p\) = mass × velocity). We can use this relationship to find the speed of the projectile before the collision: $$mc = m \times v_{projectile}$$ Divide both sides by \(m\) to find \(v_{projectile}\): $$v_{projectile} = c$$ So the speed of the projectile before the collision is \(c\).

Find the Mass of The New Particle

When the two particles collide and form a new single particle, the mass of this new particle (\(m_{new}\)) will be the sum of the masses of the initial particles: $$m_{new} = m + 2\sqrt{2}m$$ Factoring out \(m\), we have: $$m_{new} = m(1 + 2\sqrt{2})$$

Determine the Speed of the New Particle After the Collision

Using the conservation of momentum, the momentum before the collision is equal to the momentum after the collision. Let the final speed of the new particle be \(v_{new}\), then: $$\text{Initial momentum} = \text{Final momentum}$$ We have the initial momentum as \(mc\), and the final momentum as \(m_{new} \times v_{new}\): $$mc = m_{new} \times v_{new}$$ Substitute the expression for \(m_{new}\) from Step 3: $$mc = m(1 + 2\sqrt{2}) \times v_{new}$$ To find \(v_{new}\), we can divide both sides by \(m(1 + 2\sqrt{2})\): $$v_{new} = \frac{mc}{m(1 + 2\sqrt{2})}$$ Canceling \(m\) from the numerator and denominator, we get: $$v_{new} = \frac{c}{1 + 2\sqrt{2}}$$ Now we have the solutions for each part: a) The speed of the projectile before the collision is \(c\). b) The mass of the new particle is \(m(1 + 2\sqrt{2})\). c) The speed of the new particle after the collision is \(\frac{c}{1 + 2\sqrt{2}}\).

Key concepts

Elementary Particle Experiment

In the fascinating world of physics, an elementary particle experiment involves studying the properties and interactions of the smallest known building blocks of matter. These experiments are crucial for scientists to understand the fundamental forces and constituents of the universe.

During such experiments, high-energy collisions between particles are produced in particle accelerators. Through these collisions, scientists can observe the behavior of particles under extreme conditions, create new particles, and test predictions of theoretical physics. In our given problem, the collision between a projectile particle and a target particle leads to the formation of a new particle. By analyzing the collision's outcomes, we gain insights into the mass and speed of the involved particles, which are essential parameters in the study of particle physics.

Conservation of Momentum

The conservation of momentum is a fundamental concept in physics, stating that in a closed system with no external forces, the total momentum remains constant. This principle is rooted in Newton's third law which implies that for every action, there is an equal and opposite reaction.

In our collision scenario, despite the complexity of particle interactions, the total momentum of the system before and after the collision must be the same. This law is instrumental in solving the problem at hand by allowing us to equate the initial momentum of the projectile with the final momentum of the combined particle post-collision. It's the cornerstone for understanding and calculating the behavior of objects during interactions.

Speed of Projectile

The speed of a projectile in physics refers to how fast a moving object is traveling along its path. It's an important aspect of dynamics, especially in projectile motion and collision problems.

In our exercise, it's the initial velocity of the particle before impact. To determine this speed, we apply the known values of the particle's momentum and mass. By doing so, we realize that the speed of the projectile is a crucial element in predicting the outcome of the collision, as it directly influences the final momentum and therefore the energy involved in the interaction. Understanding the speed of projectiles allows us to calculate trajectories, forces, and post-collision velocities.

Mass of Particles

The mass of an object is a fundamental property in physics, representing the amount of matter it contains. In particle physics, the mass of particles is particularly significant as it contributes to the object's inertia and gravity.

When we deal with collisions, as in our exercise, the masses of the colliding objects play a critical role. They determine how momentum is shared between objects in a collision and influence how much energy is transferred or transformed in the process. In the context of our problem, both the mass of the initial projectile and the target particle must be considered to ascertain the mass of the new particle formed after the collision. It's an excellent example to illustrate how mass is conserved in physical interactions, despite the transformation of particles.