Problem 10 - Advanced Engineering Mathematics

Question

Suppose a parlicle movesin space so that \(\mathbf{a}(t)=\mathbf{0}\) for all time \(t\). Describe its path.

Step-by-Step Solution

Answer

The path is a straight line with constant velocity.

Step 1: Identifying the Acceleration

First, let's understand the given information: the acceleration vector, denoted as \( \mathbf{a}(t) \), of a particle is given as \( \mathbf{0} \) for all time \( t \). This means that the particle experiences no acceleration at any point in time.

Step 2: Understanding Physical Implications

If the particle's acceleration is zero, it means there are no net external forces causing the particle to speed up, slow down, or change direction. As a result, according to Newton's First Law of Motion, the particle will continue its current state of motion.

Step 3: Conclusion Based on Velocity

If \( \mathbf{a}(t) = \mathbf{0} \), the velocity \( \mathbf{v}(t) \) of the particle is constant. This implies that the particle is either at rest or moving at a constant velocity in a straight line. Without any further context, we can say the motion is linear and uniform.

Key Concepts

Acceleration

Acceleration is a concept that describes how the velocity of an object changes over time. It can be visualized as how quickly something speeds up or slows down.
In mathematical terms, acceleration is the derivative of velocity with respect to time. Think about when you push down the gas pedal in a car — you're accelerating, increasing your speed. If you let off the gas pedal or hit the brakes, your speed decreases, and the car is decelerating. Acceleration provides information about how fast a change in speed occurs:

• Positive acceleration means speeding up.
• Negative acceleration, often called deceleration, means slowing down.
• No acceleration means maintaining a constant speed.

For the exercise given, since the acceleration is zero, the particle's velocity doesn’t change. This unchanging velocity points towards a steady, consistent movement with no ups and downs.

Velocity

Velocity is the speed of an object in a given direction. Unlike speed, which is only scalar, velocity is a vector, which means it has both magnitude and direction.
Imagine you're driving north at 60 miles per hour — that's your velocity because it specifies how fast and in which direction you're traveling. The relationship between velocity and acceleration is crucial:

• If acceleration is zero, velocity stays the same.
• If acceleration is positive, velocity increases.
• If acceleration is negative, velocity decreases.

In the original exercise, the particle's velocity does not change because its acceleration is zero. This means, essentially, the particle's motion is uniform. It travels in a straight path at a constant speed. Whether it's moving, or staying still, without acceleration, its state of velocity remains."

Newton's First Law

Newton's First Law, often called the Law of Inertia, states that an object will remain at rest or in uniform motion in a straight line unless acted upon by an external force.
This foundational principle basically tells us how movement occurs in the absence of forces. To truly understand this, picture space as an endless ice rink. If you slide a puck, it keeps sliding in a straight line indefinitely, unless a force (like friction or a player's stick) alters its motion.
In the given scenario, this law explains why the particle continues its motion unaffected:

• No acceleration means no net force.
• The particle stays in uniform motion as nothing alters its state.

Without outside interference, velocity persists, proving that unless there's a force applied, change won't occur.