Dynamics Lecture 02: Particle kinematics, Rectilinear continuous motion part 1
Summary
TLDRThis 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
📚 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.
Mindmap
Keywords
💡Particle Kinematics
💡Rectilinear Motion
💡Position Vector
💡Velocity
💡Acceleration
💡Displacement
💡Instantaneous Velocity
💡Speed
💡Distance Traveled
💡Continuous Motion
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
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