Essential University Physics Richard Wolfson

2nd Edition

Chapter 3 Lecture

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第三章 Motion in Two and Three Dimensions ⼆二維與三維的運動

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In this lecture you’ll learn 本章內容簡介 •  To describe position, velocity, and

acceleration in three-dimensional space using the language of vectors用向量來表示位置 速度 加速度

•  To manipulate vectors algebraically

如何計算向量 •  To transform velocities to different

reference frames速度在相對的（或不同的）坐標的轉換

•  To solve problems involving constant acceleration in two dimensions解二維等加速度的問題 –  Including projectile motion due to the

constant acceleration of gravity near Earth’s surface包括地表的拋體問題

•  To evaluate acceleration in circular motion 計算圓周運動的加速度

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Vectors 向量 •  A vector is a quantity that has both magnitude and

direction.向量有大小（只有正數）有方向（分：東南西北上下） –  In two dimensions it takes two numbers to specify a vector.

在二維空間用2個數（有正有負）表示向量 –  In three dimensions it takes three numbers.

在三維空間用3個數（有正有負）表示向量 –  A vector can be represented by an arrow whose length

corresponds to the vector’s magnitude.

向量也可以用一隻箭表示： 箭長為向量的大小 箭頭指的方向為向量的方向 Note: 所以向量有兩種表示的方法; 第一種用來做計算, 第二種是以示意為目的的.

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Position is a vector quantity.位置是向量

–  In 2D, an object’s position is specified by giving its distance from an origin and its direction relative to an axis.

二維的情形：一物體的位置由相對原點的距離與相對其一軸的方向來定 – 例如： A position vector r1

describes a point 2.0 m from the origin at a 30˚ angle to the axis.

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Adding Vectors 向量的加法

用圖表示兩個向量(1,2)的相加 •  To add vectors graphically, place the tail of the first vector at

the head of the second. 1的尾接2的頭 –  Their sum is then the vector from the tail of the first

vector to the head of the second. 第1個的頭到第2個的尾是相加的結果=向量3 1,2,3三向量形成一三角形

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•  To subtract vectors, add the negative of the second vector to the first.

•  Vector arithmetic is commutative and associative:

Vector Arithmetic 向量的代數 •  To multiply a vector by a scalar, multiply the

vector’s magnitude by the scalar. 乘一數 –  For a positive scalar the direction is unchanged.乘一正數 –  For a negative scalar the direction reverses.乘一負數

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Unit Vectors 單位向量 •  Unit vectors have magnitude 1, no units, and point

along the coordinate axes.

–  They’re used to specify direction in compact mathematical representations of vectors.

–  Unit vectors in the x, y, and z directions are designated ˆˆ ˆ, , and .i j k

–  Any vector in two dimensions can be written as a linear combination of 線性組合成任意向量

–  Any vector in three dimensions can be written as a linear combination of

ˆ ˆ and .i j

ˆˆ ˆ, , and .i j k

ˆˆ ˆ, , and .i j k

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Vector Arithmetic with Unit Vectors 使用單位向量做計算

•  To add vectors, add the individual components: –  , then

–  Similarly,

If A = Axi + Ay j and

B = Bxi + By j

A+B = Ax + Bx( ) i + Ay + By( ) j

•  To multiply by a scalar, distribute the scalar; that is, multiply the individual components by the scalar:

–  – 

If A = Axi + Ay j

then cA = cAxi + cAy j

A−B = Ax − Bx( ) i + Ay − By( ) j

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Velocity and Acceleration Vectors

•  Velocity is the rate of change of position. –  The average velocity over a time interval ∆t is the change

in the position vector divided by the time. •  Here dividing by ∆t means multiplying by the scalar 1/∆t:

•  Instantaneous velocity is the time derivative of position:

•  Acceleration is the rate of change of velocity:

v = ΔrΔt

v = limΔt→0

ΔrΔt

=drdt

Average: a = Δ

vΔt

Instantaneous: a = d

vdt

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Velocity and Acceleration in Two Dimensions •  An acceleration acting for time ∆t produces a

velocity change 在∆t時間內加速度對速度的改變 –  The change adds vectorially to give the new velocity:

aΔv = aΔt.

v = v0 +aΔt

•  The new velocity depends on the magnitude of the acceleration as well as its direction:

In general:both speed and direction change

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Relative Motion 相對運動 •  An object moves with velocity relative to one frame of

reference.一物體相對一座標的速度為v‘ •  That frame moves at relative to a second reference

frame. 此座標相對於另一座標速度為Ｖ •  Then the velocity of the object relative to the second

frame is 此物體相對於另一座標的速度是v

!v

V

v = !v +V .

•  Example: 1-2, 2-3 => 1-3=1-2+2-3 –  A jetliner flies at 960 km/h relative to

the air, heading northward. There’s a wind blowing eastward at 190 km/h. In what direction should the plane fly?

–  The vector diagram identifies the quantities in the equation, and shows that the angle is 11°.

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Constant Acceleration 等加速度

•  With constant acceleration, the equations for one-dimensional motion apply independently in each direction.2D: –  The equations take a compact form in vector notation. –  Each equation stands for two or three separate equations. 加速度為已知, 可得位置與速度:

–  For example, in two dimensions, the x- and y-components of the position vector can be written as 上之第2式可以分開 成x,y兩式：

v = v0 +a t −−−−−−− (1)

r = r0 +v0t + 1

2

at 2 −−− (2)

r21

0 0 221

0 0 2

x x

y y

x x v t a t

y y v t a t

= + +

= + +

a = axi + ay j

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Projectile Motion 拋體運動 •  Motion under the influence of gravity near Earth’s surface

has essentially constant acceleration whose magnitude is g = 9.8 m/s2, and whose direction is downward.

Such motion is called projectile motion.

vx = vx0

vy = vy0 − gt

x = x0 + vx0t

y = y0 + vy0t −12 gt2

g

–  Equations for projectile motion, in a coordinate system with y axis vertically upward:

–  Horizontal and vertical motions are independent:

ax = 0,ay = −g

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Projectile Trajectories 拋物體的軌跡線 •  The trajectory of an object in projectile motion is a

parabola, unless the object has no horizontal component of motion.拋物線或直線 –  Horizontal motion is

unchanged, while vertical motion undergoes downward acceleration:

–  Equation for the trajectory:

y = x tanθ0 −

g2v0

2 cos2θ0

x2

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Uniform Circular Motion 等速率圓周運動 •  When an object moves in a circular path of radius r at

constant speed v, its acceleration has magnitude 等速率（半徑r)的圓周運動,

加速度大小為一定值：

–  The acceleration vector points toward the center of the circle.加速度的方向是向心的, 與位置方向相反與速度方向垂直

–  Since the direction of the acceleration keeps changing, this is not constant acceleration.非等加速度

a = v2

r

–  Constant acceleration in two dimensions implies a parabolic trajectory. 如上, 2D的等加速度運動的軌跡線是拋物線

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•  is perpendicular to while is tangential to .

–  The figure shows a car braking as it rounds a curve.

Nonuniform Circular Motion非等速率圓周運動 •  In nonuniform circular motion, speed and path

radius can both change.速率與圓半徑都可變 •  The acceleration has both radial and tangential

components: a = ar +at

arvatv

2 2t ra a a= +

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Summary •  In two and three dimensions, position, velocity, and

acceleration become vector quantities. –  Velocity is the rate of change of position:

–  Acceleration is the rate of change of velocity:

v = drdta = d

vdt

•  In general, acceleration changes both the magnitude and direction of the velocity.

•  Projectile motion results from the acceleration of gravity.

•  In uniform circular motion, the acceleration has magnitude v2/r and points toward the center of the circular path.

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•  Homework : 16, 26,27,32,38,44,48,64,70,