Problem 11 - University Physics with Modern Physics

Question

In an experiment in space, one proton is held fixed and another proton is released from rest a distance of 2.50 \(\mathrm{mm}\) away. (a) What is the initial acceleration of the proton after it is released? (b) Sketch qualitative (no numberst) acceleration-time and velocity-time graphs of the released proton's motion.

Step-by-Step Solution

Answer

(a) Calculate the force and use Newton's second law to find acceleration. (b) Acceleration decreases over time, velocity increases.

Step 1: Understanding the Problem

We need to find the initial acceleration of a proton released in space due to the electrostatic repulsion from a nearby proton. The distance between them is given as 2.50 mm.

Step 2: Apply Coulomb's Law

First, we apply Coulomb's law to determine the electrostatic force between the two protons. Coulomb's law is given by \[ F = k \frac{|q_1 q_2|}{r^2} \]where \( k = 8.99 \times 10^9 \, \mathrm{N \, m^2/C^2}\) (Coulomb's constant), \( q_1 \) and \( q_2 \) are the charges of the protons (each \( 1.6 \times 10^{-19} \, C \)), and \( r = 2.50 \, \mathrm{mm} = 2.50 \times 10^{-3} \, \mathrm{m} \).

Step 3: Calculate the Force

Plug the values into Coulomb's law: \[ F = (8.99 \times 10^9) \frac{(1.6 \times 10^{-19})^2}{(2.50 \times 10^{-3})^2} \]Calculate this expression to find the force.

Step 4: Compute the Acceleration

To find the acceleration, use Newton's second law: \[ a = \frac{F}{m} \]The mass \( m \) of a proton is \( 1.67 \times 10^{-27} \, \mathrm{kg}\).Substitute the force calculated earlier to find the initial acceleration \( a \).

Step 5: Sketching Acceleration-Time Graph

Qualitatively, the acceleration of the proton will decrease over time as the distance increases, due to the inverse square nature of the force. The graph starts high and slopes downward.

Step 6: Sketching Velocity-Time Graph

Initially, the velocity of the proton is zero, and it increases over time due to the repulsive force. The velocity-time graph will start at the origin and rise upwards.

Key Concepts

Coulomb's Law

Coulomb's Law is fundamental in electrostatics. It describes the force between two charged particles. The law states that the electrostatic force between two charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them. This force can be calculated using the formula \( F = k \frac{|q_1 q_2|}{r^2} \), where:

• \( F \) is the electrostatic force,
• \( k = 8.99 \times 10^9 \, \mathrm{N \, m^2/C^2} \) (Coulomb's constant),
• \( q_1 \) and \( q_2 \) are the charges of the particles,
• \( r \) is the distance between the charges.

In the context of protons, since they both have the same positive charge, they repel each other with a force calculated using this law. This principle is crucial for understanding the initial motion of particles in electrostatic fields.

Proton motion

Imagine two protons situated in a vacuum. One is stationary, while the other is initially released. The released proton experiences a repulsive force due to its like charge with the stationary proton. This force accelerates the proton away from its initial position.
The motion of the proton is initially dominated by this electrostatic repulsion. As it accelerates, the distance between the protons increases. This greater distance causes the repulsive force, and thus the acceleration, to decrease over time due to the inverse-square nature of Coulomb's Law. It is crucial to understand this behavior as it helps predict how charged particles move in electrostatic fields.

Acceleration-time graph

An acceleration-time graph visualizes how the acceleration of a particle like our proton changes over time. At the very start, right after release, the acceleration is at its peak due to the close proximity of the two protons. This is because the electrostatic force, which causes acceleration, is strongest at shorter distances.
Over time, as the proton moves away, the acceleration decreases. This decrease happens because the force diminishes with increasing distance, following the inverse-square nature of Coulomb's Law. In a qualitative sense, this graph will show a downward slope, starting high and gradually declining as time progresses.

• Initially steep, reflecting strong acceleration.
• Gradually slopes downward as distance increases.
• Never reaches zero, but approaches it as protons move apart significantly.

Understanding this graph helps in grasping how forces affect motion over time.

Velocity-time graph

The velocity-time graph is key to understanding how the speed of the released proton changes. At the time of release, the proton's velocity is zero since it starts from rest. As it gains speed due to the repulsive force, the velocity increases.
Over time, even though the acceleration decreases, the velocity of the proton continues to rise because it keeps getting pushed, albeit more gently as distance increases. This results in a graph that starts at the origin (velocity zero) and rises sharply initially, then continues to rise at a slower pace.

• Begins at zero, indicating initial rest state.
• Rises rapidly due to initial high acceleration.
• Slope becomes less steep over time as acceleration decreases.

This graph visually represents the cumulative impact of continuous force application on particle velocity.