Question

True or False The velocity and speed of the particle are the same at \(t=\pi / 4 .\) Justify your answer.

Short answer

Insufficient information to determine if the velocity and speed of the particle are the same at \(t= \pi / 4\)

Step-by-step solution

Identifying Information available

The given exercise does not provide enough information about the movement of the particle at \(t = \pi/4\). Therefore, it is crucial to note that without substantial information, such as graphical representation or mathematical equations related to the particle’s movement, we cannot determine the velocity and speed of the particle at this particular point in time.

Understanding Velocity vs Speed

The major point here is comprehending that the speed of a particle refers strictly to how fast a particle is moving while the velocity refers to the speed of the particle in a specific direction. So, if we were given more information about the movement of the particle at \(t = \pi/4\), we could compare the speed and velocity. However, due to lack of such information, we can’t definitively state whether they’re the same or different.

Answer Justification

Based on the provided information, or lack thereof, no conclusive answer can be drawn whether the velocity and speed of the particle are the same at \(t = \pi/4\). If we had further knowledge on the particle’s movement, we could answer this question by determining the magnitude of the velocity (which would represent the speed) and the direction (which forms part of the velocity).

Key concepts

Particle Movement

When we talk about particle movement, it's essential to understand that this refers to how a particle changes its position over time. In physics, this is often analyzed by studying the motion graphs or equations of motion that describe the path of the particle. Particle movement can be linear, circular, or even random, depending on the forces acting upon the particle.

For instance, if a particle is moving in a straight line at a constant speed, its movement is quite straightforward to comprehend. On the other hand, if it's moving in a circle, like a planet orbiting the sun, the movement gets more complex due to the continually changing direction of motion, which is crucial for understanding velocity.

Magnitude and Direction

In physics, magnitude refers to the size or quantity of a physical quantity, without considering its direction. When discussing motion, magnitude often relates to speed, which indicates how fast an object is moving. It is a scalar quantity, meaning it only has magnitude and no direction.

On the flip side, when we add direction to the mix, we're dealing with vector quantities. For example, while speed might tell us that a car is traveling at 60 mph, velocity tells us that the car is moving at 60 mph to the north. This combination of magnitude and direction is pivotal because it provides a complete description of the motion, allowing us to predict future positions of the particle based on its current state. Understanding both magnitude and direction is essential to completing a comprehensive study of kinematics in physics.

Velocity and Speed Distinction

To discern the velocity and speed distinction, it's important to recognize speed as a measure of how fast an object is moving regardless of its travel direction, while velocity includes both the speed and the direction of the object's movement. As such, speed is a scalar quantity, and velocity is a vector quantity.

For instance, if two cars are traveling at 50 mph, one east and the other west, their speeds are the same, but their velocities are different due to their opposite directions. This distinction is crucial in solving problems related to particle movement because it helps in understanding that velocity can be negative or positive—indicating direction—while speed is always a non-negative value. Therefore, unless the direction of a particle's movement is known as well as its speed, we cannot fully describe its velocity.