## 2. motion in one dimension

2-1: Position, Velocity, and Speed

Position: is the location of the particle with respect to a chosen reference point that we can consider to be the origin of a coordinate system.

Displacement: the displacement  $\Delta x$  of a particle is defined as its change in position in some time interval. As the particale moves from an initial position  $x_{i}$  to a final position  $x_{f}$  its displacement is given by:

 $\Delta x\equiv x_{f}-x_{i}$   (2-1)

Distance: is the length of a path followed by a particle. distance is always represented as a positive number, whereas displacement can be either positive or negatve.

Average velocity$\nu _{x,avg}$  of a particle is defined as the particle’s displacement  $\Delta x$  divided by the time interval  $\Delta t$  during which that displacement occus:

 $\nu _{x,avg}\equiv \frac{\Delta x}{\Delta t}$   (2-2)

Average speed:  $\nu _{avg}$  of a partical, a scalar quantity, is defined as the total distance  $d$  traveled divided by the total time interval required to travel that distance:

 $\nu _{avg}\equiv \frac{d}{\Delta t}$  (2-3)

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2-2: Instantaneous velocity and speed

Instantaneous velocity:  $v_{_{x}}$  equals the limiting value of the ratio  $\frac{\Delta x}{\Delta t}$  as  $\Delta t$  approaches zero:

$\lim_{\Delta t\rightarrow 0} \frac{\Delta x}{\Delta t}$   (2.4)

In calculus notation, this limit is called the derivative of  $x$  with respect to  $t$  , written $\frac{dx}{dt}$:

 $v_{x}\equiv \lim_{\Delta t\rightarrow 0 }\frac{\Delta x}{\Delta t}=\frac{dx}{dt}$  (2.5)

Instantaneous velocity can be positive, negative, or zero.

Instantaneous speed: of a particle is defined as the magnitude of its instantaneous velocity.

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2-3 : Analysis Model: Particle Under Constant Velocity

$v_{x}=\frac{\Delta x}{\Delta t}$  (2.6)

Remembering that  $\Delta x=x_{f}-x_{i}$  ,we see that   $v_{x}=\frac{x_{f}-x_{i}}{\Delta t}$  , or  $x_{f}=x_{i}+v_{x}\Delta t$

Position as function of time for the particle under constant velocity model

 $x_{f}=x_{i}+v_{x} t$ ( for constant  $v_{x}$)   (2.7)

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2-4: Acceleration.

when the velocity of a particle changes with time, the particle is said to be accelerating.

– The Average acceleration  $a_{x,avg}$  of the particle is definded as the change in velocity  $\Delta v_{x}$  divided by the time interval  $\Delta t$  during which that change occurs:

$a_{x,avg}\equiv \frac{\Delta v_{x}}{\Delta t}=\frac{v_{xf} - v_{xi}}{t_{f}-t_{i}}$

Instantaneous acceleration: it is useful to define it as the limit of the average acceleration as  $\Delta t$  approaches zero.

$a_{x}\equiv \lim_{\Delta t\rightarrow 0}\frac{\Delta v_{x}}{\Delta t}=\frac{dv_{x}}{dt}$

the instantaneous acceleration equals the derivative of the velocity  with respect to time

$a_{x}=\frac{dv_{x}}{dt}=\frac{d}{dt}(\frac{dx}{dt})=\frac{d^{2}x}{dt^{2}}$

that is, in one-dimensional motion, the acceleration equals the second derivative of  $x$  with respect to time.

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2-6: Analysis Model: Particle Under Constant Acceleration

the particle under constant acceleration. we generate several equations that describe the motion of a particle for this model.

$a_{x}=\frac{v_{xf}-v_{xi}}{t-0}$

or:

$v_{xf}=v_{xi}+a_{x}t$  (for constant  $a_{x}$  )  (2.13)

Becouse velocity at cnstant acceleration varies linearly in time according to Equation (2.13), we can express the average velocity in any time interval as the arithmetic mean of the initial velocity  $v_{xi}$  and the final velocity  $v_{xf}$  :

$v_{x,avg}=\frac{v_{xi}+v_{xf}}{2}$  (for constant  $a_{x}$  )  (2.14)

Notice: that this expression for average velocity applies only in situations in which the acceleration is constant.

Position as a function of velocity and time for the particle under constant acceleration model

$x_{f}=x_{i}+\frac{1}{2}(v_{xi}+v_{xf})t$  (for constant  $a_{x}$  )  (2.15)

Position as a function of time for the particle under constant acceleration model

$x_{f}=x_{i}+v_{xi}t+\frac{1}{2}a_{x}t^{2}$  (for constant  $a_{x}$  )  (2.16)

Position as a function of position for the particle under constant acceleration model

$(v_{xf})^{2} =(v_{xi})^{2} +2a_{x}(x_{f}-x_{i})$  (for constant  $a_{x}$  )  (2.17)

Table:Kinematic Equations for motion of a Particle Under Constant Acceleration

 Information given by Equation Equation Equation Number Velocity as a function of time $v_{xf}=v_{xi}+a_{x}t$ 2.13 Position as a function of velocity and time $x_{f}=x_{i}+\frac{1}{2}(v_{xi}+v_{xf})t$ 2.15 Position as a function of time $x_{f}=x_{i}+v_{xi}t+\frac{1}{2}a_{x}t^{2}$ 2.16 Velocity as a function of position $(v_{xf})^{2} =(v_{xi})^{2} +2a_{x}(x_{f}-x_{i})$ 2.17

Note: Motion is along the x axis

Remember: thet these equations of kinematics cannot be used in a situation in which the acceleration varies with time.

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2-7: Freely Falling Object

In the absence of air resistance, all objects dropped near the Earth’s surface fall toward the Earth with the same constant acceleration under the influence of the Earth’s gravity.

When we use the expression freely falling object, we do not necessarily refer to an object dropped from rest. a freely falling object is any object moving freely under the influence of gravity alone, regardless of its initial motion.

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Physics for Scientists and Engineers with Modern.8E