## 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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