01kinematics1l

• Speed v
The speed of a particle always equals the magnitude of its velocity at any time. It is a scalar. The speed of a particle equals to the derivative of the distance traveled with respect to time.
cos 2 α + cos 2 β + cos 2 γ = 1
For a plane polar system
r ˆ r = rer
ˆ eθ
P (r ,θ )
r
ˆ er
O
θ
2. Motion equation
x = x (t )
v v r = r (t )
y = y(t )
z = z (t )
Example The motion equation of a particle is given by
Chapter 2 Describing Motion: Kinematics in Two and Three Dimensions 2.1 Position Vector
1. Definition
r r
y
r r
O z
P（x，y，z） x
The position vector of a particle is a vector drawn from the origin of a reference frame to the xyz position of the particle.
r r r r ( t + Δt ) − r ( t ) v = lim Δt → 0 Δt r r r Δ r dr v = lim = Δt → 0 Δ t dt
z
P1
r v1
v Δr
P2
r v2
Unit: m/s O y x Direction The direction of the instantaneous velocity at any point in a particle’s path is along a line tangent to the path at that point and in the direction of motion. The instantaneous velocity is the derivative of the position vector with respect to time.
r π ˆ π ˆ r ( t ) = 3 sin t i + 3 cos t j (m). 6 6
The path of the particle can be determined. y t=0 t =9s O t =3s t =6s x
π x ( t ) = 3 sin t 6 π y( t ) = 3 cos t 6
r r Δr v = Δt
Unit: m/s
Direction：it points in the direction of the displacement. y A
v rA
z o
v Δr
v rB
B
x
r • Instantaneous Velocity v
The instantaneous velocity is defined as the limit of the average velocity as the time interval Δt approaches zero.
The subjects that occupied physical scientists through the end of the nineteenth century----mechanics, light, heat, sound, electricity and magnetism-------are usually referred to as classical physics.
y
• We will work with two coordinate systems: – Cartesian (x, y,z)
– Plane polar (r, θ ) z
P（x，y，z）
ˆ j
ˆ k O
ˆ i
x
• Note that the x, y ,z and θ
values can be negative as well as positive. Unit vectors
Remark
1.We need to make a distinction between the displacement and the distance. Δs ( scalar) 2. Δ t → 0 y
r Δr vector
v ds = dr
A ΔS B
v rA
O z
v Δrv
rB
x
3.
y
A
v rA
z O
v Δr
v rB
B
x
Displacement is the change in position of the object. It equals the difference between its final position vector and initial position vector.
v v v Δr = r ( t + Δt ) − r ( t )
The displacement is independent on the choice of the origin.
For a Cartesian system:
v v v Δr = r ( t + Δt ) − r ( t )
Course Textbook
Physics For Scientist and Engineers with Modern Physics (Third Edition) Douglas C. Giancoli Higher Education Press
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v ˆ Δr = Δxi + Δyˆ j
v Δr = (Δx )2 + (Δy )2
Δx = x ( t + Δ t ) − x ( t ) Δy = y ( t + Δt ) − y ( t )
y
Δy tan θ = Δx
v rA
O
A
v Δr
B
v rB
xA
θ
Δy
Δx
xB x
θ is measured with respect to the x axis.
P (r ,θ )
r
ˆ eθ
O
ˆ er
θ
For a Cartesian system:
y
P (x，y，z)
v ˆ ˆ r = x i + yˆ + z k j
v r = r = x2 + y2 + z2
β γ
O y
ˆ j
Baidu Nhomakorabeaˆ i
α
x
x cos α = v r y cos β = v r z cos γ = v r
v Δr and Δr
v Δr ≠ Δr
2. Velocity
Velocity is the rate at which the position changes. • Average Velocity The average velocity of a particle during the time interval Δt is defined as the ratio of the displacement to the time interval.
Particle (Point Mass质点)： An object whose position can be described by a single point. Anything can be considered to be a particle – a molecule, a person or a galaxy – as long as we can reasonably ignore its internal structure. It is a model in mechanics. Particle (point mass), Rigid body reference frame (参照系)： To describe the position of an object, the other object which is referred to (Reference Frame ）should be chosen. It is arbitrary. coordinate system (坐标系)： coordinate systems that are attached to reference frames.
z α is the angle between the position vector and the x axis.
β is the angle between the
position vector and the y axis.
z
ˆ k
O
x
γ is the angle between the
position vector and the z axis.
v r dr v = v = dt v v Δr Δr = lim = lim Δt → 0 Δ t Δt → 0 Δ t
Questions
1. What are fundamental SI Units? 2. What is the Scientific notation? 3. How do you find dimensions of quantities?
Mechanics
The study of the motion of objects, and the related concepts of force and energy. Kinematics (运动学)：the mathematical methods of describing motion. Dynamics(动力学)：the study both of motion and the forces that bring about the motion.It deals with force and why objects move as they do. Static Equilibrium (静力学)：study of the conditions necessary for static equilibrium to exist.
It is classical physics we must master to understand the macroscopic world we live in, and classical physics is the main subject of the first part of the course.
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Physics is a science of matter and energy, and includes the principles that govern the motion of particles and waves, the interaction of particles, and the properties of molecules , atoms and atomic nuclei , as well as larger-scaled systems, such as gases, liquids, and solids.
x 2 + y 2 = 32
Trajectory equation
f ( x, y, z)=0
2 .2 Displacement , Velocity v 1. Displacement Δr v rA at time t v rB at time t + Δt
Definition-Displacement