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Kinematics Equations

Various representations are used to represent motion of any object like pictorial representation, graphs, and verbal representation etc. The branch of mechanics which studied the motion of an object is called kinematics. It describes the concept of motion with graphical representation of its basic term like displacement, velocity, speed, acceleration etc. and motion equation which gives the relation between these basic terms. It is mainly given the description of relative positions and changes in the position of an object with respect to time. 

Here we discuss about four kinematic equations. These are used to detect the unknown value or variable with the use of known information or variables. These equations describe motion either at constant velocity or at constant acceleration. The time period in which the acceleration is changed is not used in these equations. Now we discuss all four kinematic equations and their uses with doing some problem on this topic.

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What is Kinematics?

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In Classical mechanics, we basically define motion in terms of space and time, and ignore the agents that cause that motion. This portion of the classical mechanics is called kinematics.

From our everyday experience, we identify that motion represents a continuous change in the location of an object.
There are three types of motion:

a) Translational Motion: A car moving on a highway is an example of translational motion.

b) Rotational Motion: The Earth’s revolve on its axis, is an example of rotational motion.

c) Vibrational Motion: The motion of the pendulum is an example of vibrational motion. 
We could also say that Kinematics is the study of objects in motion.The main concepts of Kinematics include speed, velocity, acceleration, time, distance and displacement.
1D Kinematics Equations are as follows:
$d$ = $V_{i}\ t\ +$ $\frac{1}{2}$ $\times\ a\ \times\ t^{2}$

$V_{f}$ = $V_{i}\ +\ a\ \times\ t$

$V_{f}^{2}$ = $V_{i}^{2}\ +\ 2ad$.
Where,

$d$ is displacement and $d$ = $\frac{V_{f}\ +\ V_{i}}{2}$,

$a$ is acceleration,

$t$ is time,

$V_f$ is final velocity, 

$V_i$ is initial velocity.

or

There are three equation of motion, which are nothing but kinematics equations, which are :

1) $v$ = $u\ +\ at$

2) $S$ = $ut\ +$ $\frac{1}{2}$ $at^2$

3) $v^2$ = $u^2\ +\ 2as$.

Where,

$v$ = Final Velocity,

$u$ = Initial velocity,

$a$ = acceleration,

$s$ = distance traveled by a body,

$t$ = time taken.
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Inverse Kinematics

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Kinematics can be stated as the procedure to calculate the position of the end of a linked structure, if angle of all the joints is already known. Kinematics is easy as there is one solution.

Inverse Kinematics does the reverse of kinematics and in case we have the end point of a particular structure, certain angle values would be needed by the joints to achieve that end point. It is a little difficult and has generally more than one or even infinite solutions.

Kinetics Vs Kinematics

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Below are given the Comparison between Kinetics and Kinematics:

Kinetics
Kinematics
  • Kinetics is a concept of the classical mechanics and it deals with the motion of various objects and the forces which act on the objects either they are in rest or in motion.
  • Here the force for cause of the motion is studied.
  • Kinematics on the other hand, only deals with the motion or movement of various objects, and does not focus on the forces which causes the motion.
  • Here the force for the cause of the motion is not studied.

4 Kinematics Equations

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The Kinematics Equations are as follows:
$V$ = $V_o\ +\ at$

$X\ -\ Xo$ = $V_o\ t\ +$ $\frac{1}{2}$ $a\ t^{2}$

$V^{2}$ = $V_o^{2}\ +\ 2a(X\ -\ X_o)$    

$X\ -\ X_o$ = $\frac{1}{2}$ $(V_o\ +\ V)t$.
Where, 

$V$ is final velocity $(m/s)$,

$V_o$ is initial velocity $(m/s)$, 

$a$ is acceleration $(m/s^2)$,

$t$ is time $(s)$,

$X$ is final displacement $(m)$, 

$X_o$ is initial displacement.

More than one unknowns could be solved for each other by using more than one equation. If any equation needs to be solved for two components, we need to find a common piece (example time), solve for this piece for the rest of the unknown components and set them equal to each other to solve for the other unknowns. It is easier to learn the equations as:
$d_f\ -\ d_i$ = $V_i\ t\ +$ $\frac{1}{2}$ $at^2$

$V^2$ = $V_i^{2}\ +\ 2a(df - di)$

$df\ -\ di$ = $\frac{1}{2}$ $\times\ (V_i\ +\ V_f)t$.

2D Kinematics Equations

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Since it is 2D equation, we are considering $X$ and $Y$ axis. 2D equations along $x$ and $y$ direction are given below. Considering $x$-direction,
$a_x$ = constant      

$V_fx$ = $V_ix\ +\ a_x\ \Delta t$

$X_f$ = $X_i\ +\ V_ix\ \Delta t\ +$ $\frac{1}{2}$ $a_x\ \Delta t^2$

$\Delta t$ = $\frac{V_{fx} - V_{ix}}{a_{x}}$

$V_{fx}^{2}$ = $V_{ix}^{2}\ +\ 2ax\ \Delta x$.        

$X_f$ = $X_i\ +$ $\frac{1}{2}$ $(V_fx\ +\ V_ix)\ \Delta\ t$.
and Considering y-direction,
$a_y$ = constant

$V_fy$ = $V_iy\ +\ a_y\ \Delta t$

$y_f$ = $y_i\ +\ V_iy\ \Delta t\ +$ $\frac{1}{2}$ $a_x\ \Delta t^2$

$\Delta t$ = $\frac{V_{fy}\ -\ V_{iy}}{a_{y}}$

$V_{fy}^{2}$ = $V_{iy}^{2}\ +\ 2ay\ \Delta y$

$y_f$ = $y_i\ +$ $\frac{1}{2}$ $(V_fy\ +\ V_iy)\ \Delta t$.
Where,

$V_f$ is final velocity $(m/s)$,

$V_i$ is initial velocity $(m/s)$, 

$a$ is acceleration $(m/s^2)$,

$t$ is time $(s)$,

$X$ is final displacement $(m)$,

$X_0$ is initial displacement.

Projectile motion is the best example of the motion of the object in the Two dimension. Here the object has the motion in both the $x$ and the $y$ direction in the Horizontal direction and in the Vertical direction or we can say that the object has the both components of the velocity.

Kinematics Problems

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Below are the problems based on Kinematics Equations:

Solved Examples

Question 1: Initial velocity of a truck is zero and it is at rest. It experiences a uniform acceleration during the time interval of 5.21 seconds. Distance covered by the truck is 110 m. Find the acceleration?
Solution:
 
Distance traveled s = 110m,
      initial Velocity $v_{i}$ = 0,
      time taken t = 5.21 s,
      acceleration a = ?
By using the Kinematic Equation we can conclude that,
s = $v_{i}$t + $\frac{1}{2}$ a $t ^{2}$,
110 m = (0) $\times$ (5.21)+ $\frac{1}{2}$ $\times$ a $(5.21)^{2}$,
a = 8.10 m/$s^{2}$.

 

Question 2: A particle is moving in cm along the x-axis after t seconds of travel is represented by the equation x = 14t2 - t + 10. Find its average Velocity after 3s of its travel?
Solution:
 
The particles position when t = 0 is x = 10cm.
When t = 3s, x = 133 cm.
Average Velocity, Vav = $\frac{\Delta x}{\Delta t}$
                                = $\frac{133 - 10}{3 - 0}$
                                = 41 cm/s.
 

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