Dynamics Lecture 02: Particle kinematics, Rectilinear continuous motion part 1

Summary

This video delves into particle kinematics, focusing on rectilinear continuous motion. It explains fundamental concepts like position, velocity, and acceleration using a 3D coordinate system. Position is represented by a vector, while velocity and acceleration are defined as the first and second time derivatives of position, respectively. The video simplifies the discussion for rectilinear motion by using a one-dimensional axis. It distinguishes between displacement, velocity, and speed, and clarifies the difference between distance traveled and displacement, using examples to illustrate these concepts.

Takeaways

• 📏 The position of a particle is represented by a position vector drawn from the origin to the particle in a 3D XYZ rectangular coordinate system.
• 🔄 The position vector changes as the particle moves, with new coordinates represented by a new vector.
• 📉 Displacement is the vector difference between the initial and final position vectors of a particle.
• 🚀 Velocity is the rate of change of position over time, calculated as the time derivative of the position function.
• 🌪 Acceleration is the rate of change of velocity, or the second time derivative of the position function.
• 📈 Both velocity and acceleration are vector quantities and functions of time.
• 🛤️ Rectilinear motion implies that the path of the particle's motion is a straight line, simplifying the coordinate system to one dimension.
• 🔢 In rectilinear motion, position is represented by a one-dimensional vector along the path, with negative values indicating the opposite direction.
• 🕒 Average velocity is defined as the change in position over a time interval, and it becomes instantaneous velocity as the time interval approaches zero.
• 🏃‍♂️ Speed is the magnitude of velocity and is a scalar quantity without direction, always non-negative.
• 🛣️ Distance traveled is the total length of the path taken by the particle, which can differ from displacement, especially in non-rectilinear motion.

Q & A

What is the general definition of position in particle kinematics?

In particle kinematics, the position of a particle is represented by a position vector, drawn from the origin of a reference coordinate system to the particle. The position vector is a function of time and is represented in Cartesian form as r = x*i + y*j + z*k, where i, j, k are the unit vectors along the x, y, and z axes, respectively.

How is velocity defined in the context of particle kinematics?

Velocity is defined as the rate of change of the particle’s position. Mathematically, it is the time derivative of the position vector, and since velocity is a vector, it also has both magnitude and direction.

What is acceleration, and how is it related to velocity and position?

Acceleration is defined as the rate of change of velocity over time. It is the time derivative of velocity and can also be expressed as the second derivative of the position vector.

What does 'rectilinear motion' refer to in particle kinematics?

Rectilinear motion refers to motion along a straight line. In this case, the position of the particle is represented along a single, one-dimensional axis, and the motion is confined to this straight path.

How is displacement defined, and how does it differ from distance traveled?

Displacement is the change in position of a particle, defined as the difference between the final and initial position vectors. Distance traveled refers to the actual length of the path covered by the particle, regardless of direction. Displacement is a vector, while distance is a scalar quantity.

What is the difference between average velocity and instantaneous velocity?

Average velocity is defined as the total displacement divided by the total time interval (Δs/Δt). Instantaneous velocity, on the other hand, is the velocity at a specific moment in time, calculated as the derivative of the position function with respect to time (ds/dt).

What is speed, and how is it related to velocity?

Speed is the magnitude of velocity and is always non-negative. Unlike velocity, which has both magnitude and direction, speed is a scalar quantity that indicates how fast an object is moving without reference to direction.

What happens to the displacement and velocity of a particle if it returns to its original position?

If a particle returns to its original position, its displacement is zero because there is no net change in position. As a result, the average velocity is also zero. However, the distance traveled may not be zero, as it represents the total path length covered.

How is rectilinear continuous motion different from erratic motion?

In rectilinear continuous motion, the position function consists of a single continuous equation over time. In contrast, erratic motion is described by a piecewise function, meaning the motion is defined by multiple equations over different time intervals.

What is the formula for calculating average speed, and how is it different from average velocity?

Average speed is calculated by dividing the total distance traveled by the total time taken, while average velocity is the displacement divided by the time interval. Unlike velocity, speed does not account for direction, and it is always positive or zero.

Outlines

00:00

📚 Introduction to Particle Kinematics

This paragraph introduces the study of particle kinematics, focusing on rectilinear continuous motion. It defines key concepts such as position, velocity, and acceleration in the context of particle motion. Position is represented by a position vector in a 3D XYZ coordinate system, with the particle's coordinates changing over time as it moves. Displacement is the vector difference between initial and final positions. Velocity is the rate of change of position over time, and acceleration is the rate of change of velocity, both of which are vector quantities. The paragraph also differentiates between rectilinear motion, which follows a straight line, and the simpler one-dimensional coordinate system used to describe it, versus more complex motion types.

Keywords

💡Particle Kinematics

Particle kinematics is the study of the motion of particles without considering the forces that cause the motion. It involves analyzing the position, velocity, and acceleration of a particle over time. In the video, the study begins with the simplest type of motion: rectilinear continuous motion, setting the foundation for understanding more complex movements.

💡Rectilinear Motion

Rectilinear motion refers to movement along a straight line. In the video, rectilinear continuous motion is discussed as the simplest form of particle motion, where the particle moves along a straight path, and its position can be described using a single dimension, simplifying the analysis of position, velocity, and acceleration.

💡Position Vector

A position vector represents the location of a particle relative to a chosen reference point, usually the origin of a coordinate system. In the video, the position vector 'r' is described as being drawn from the origin to the particle's position in a 3D coordinate system, highlighting the fundamental concept of locating a particle in space.

💡Velocity

Velocity is defined as the rate of change of a particle's position with respect to time. It is a vector quantity, meaning it has both magnitude and direction. In the context of the video, velocity is introduced as the time derivative of the position vector, providing a measure of how quickly and in which direction the particle is moving.

💡Acceleration

Acceleration is the rate of change of velocity with respect to time, indicating how quickly the velocity of a particle changes. Like velocity, acceleration is also a vector quantity. The video explains acceleration as the time derivative of velocity or the second derivative of position, highlighting its role in describing how the motion of a particle evolves over time.

💡Displacement

Displacement is the change in the position of a particle, defined as the difference between the final and initial position vectors. It is a vector quantity and is used in the video to differentiate between the initial and final positions of a particle, emphasizing its directional nature and its importance in defining motion.

💡Instantaneous Velocity

Instantaneous velocity is the velocity of a particle at a specific point in time, defined as the limit of the average velocity as the time interval approaches zero. In the video, it is described using the derivative of position with respect to time, providing a precise measurement of the particle's speed and direction at any given moment.

💡Speed

Speed is the magnitude of velocity and represents how fast a particle is moving, irrespective of direction. Unlike velocity, speed is a scalar quantity and is always non-negative. The video mentions speed to distinguish it from velocity, illustrating that it only measures the rate of motion without considering the direction.

💡Distance Traveled

Distance traveled refers to the total length of the path a particle moves along, irrespective of direction, distinguishing it from displacement. In the video, distance traveled is highlighted using an example of a ball that moves forward and returns to its starting point, showing that despite zero displacement, the distance traveled is non-zero.

💡Continuous Motion

Continuous motion refers to movement that is uninterrupted and follows a single, continuous equation over time. In the video, continuous motion is contrasted with erratic motion, which involves piecewise functions and multiple equations, emphasizing the simplicity and predictability of continuous motion in kinematic studies.

Highlights

Introduction to particle kinematics, starting with rectilinear continuous motion.

General definition of position, velocity, and acceleration in kinematics.

Position vector representation using a 3D XYZ rectangular coordinate system.

Displacement as the change in position vector.

Velocity defined as the rate of change in position over time.

Acceleration as the rate of change in velocity, or the second derivative of position.

Rectilinear motion implies motion along a straight line.

Setting up a one-dimensional axis for rectilinear motion simplifies position representation.

Position vector S indicates direction along the one-dimensional axis.

Negative S indicates a position in the opposite direction to the origin.

Continuous motion is described by a single equation for the position function.

Average velocity is the change in position over time interval.

Instantaneous velocity is the limit of average velocity as time difference approaches zero.

Speed is the magnitude of velocity and is always non-negative.

Distance traveled is the total path length and differs from displacement.

Example given: Ball traveling 5m to the right and returning to the original position.

Displacement is zero in the example, but distance traveled is 10m.

Average speed is calculated as total distance over time.

Transcripts

in this video we start the study of

particle

kinematics we will start with the

simplest topic rectilinear continuous

motion of a particle but before that I'd

like to briefly introduce the general

definition of position velocity and

acceleration if you recall from the

Statics course the position of a

particle can be represented by the

position Vector to do that we need to

first establish a reference coordinate

system in this case a 3D XYZ rectangular

coordinate

system the particle now has coordinates

of XYZ and the position of the particle

is represented by a vector r that is

drawn from the origin to this particle

this position Vector R is written in

cartian form as x i + YJ plus ZK here i

j k are the unit vectors along the x y z

AIS

respectively as long as there's motion

within the same coordinate system

position is a function of time so after

some time this particle has moved to a

new location with new coordinates X

Prime y Prime and Z Prime and its

position is now represented by a new

position Vector RP Prime the change in

position of this particle which is R

Prime minus r is defined as the

displacement as you can see displacement

is also a

vector velocity is defined as the rate

of change in the object's position from

calculus we know that it is the time

derivative function of

position acceleration is defined as the

rate of change in the object's velocity

so it is the time derivative of velocity

or the second derivative of position

note that both velocity and acceleration

are vectors and functions of

time now let's look at rectilinear

continuous

motion rectilinear means that the path

of motion is a straight

line so again we want to set up a

coordinate system so we can describe the

positions of the particle using position

vectors but since it is a rectilinear

motion we can simply set up a

onedimensional axis along the path

choose the origin to be a convenient

point and represent the positions using

vectors drawn from the origin

S1 S2 Etc since here s is a vector

negative s indicates a position in the

opposite direction to the

origin once again the position s is a

function of time here continuous motion

means that the position function s t

consists of only one equation versus

later on when we discuss erratic motion

that is when the position function is a

piecewise function that consists of

multiple

equations from time T1 to time T2 the

average velocity is defined as the

change in position over the change in

time Delta s or the displacement over

delta

T when the time difference delta T

approaches zero the average velocity

becomes the instantaneous velocity

defined as DS over DT or the time

derivative of

position sometimes we also use the word

speed which means the magnitude of

velocity and it has no sense of

direction and is always non-

negative another scalar quantity we

sometimes use is distance traveled s t

it is the length of the actual path

traveled and it is not the same as

displacement for example if this ball

travels 5 m to the right and returns to

its original position all in 5

seconds since there is no change in the

position of the ball its displacement is

zero therefore its average velocity is

also zero however its distance traveled

is a total of 10 m and therefore its

average speed is 10 M over 5 Seconds

which is 2

m/s now please answer the following

questions

for