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### 2.1 Kinematics (linear motion)

'Kinematics is the branch of mechanics concerned with the motion of objects without reference to the forces which cause the motion'.

In kinematics, we learn about both one-dimensional(linear motion) and two-dimensional motion(non linear motion). Relation between variables such as displacement, speed, velocity, and acceleration are studied here.  There are some important equation you'll have to know before you solve some exercise in the kinematics. We will go through those equations and solve some examples.

But first to the definition of various terms in kinematics.

Displacement: Displacement is a vector quantity. Displacement is the straight-line distance from the initial position to the final position. Displacement is often represented as $\Delta$x,  Delta is a greek letter used to represent a change, so $\Delta$x means "the change in x". Its unit is 'm' in MKS system.

Average Velocity
: Average velocity, $v_{avg}$ is a vector quantity. It is the ratio of the change in displacement to the change in measure of time i.e $(x_{f}-x_{i})/(t_{f}-t_{i})$  or $\Delta$x/$\Delta$t. So, as $\Delta t$ tends to zero we have the infinitesimal definition of a velocity, i.e. $v=dx/dt$. Note that $t$ represents time. Its unit is meters per second, m/s.

Average Speed: Average speed is the scalar quantity. It is the total distance traveled, not the displacement, divided by the time interval. Its unit is meters per second, m/s.

Acceleration: Acceleration is a vector quantity, which is simply a measure of change in objects velocity.  If an object's speed increases while moving in the positive direction, its acceleration is positive. If the speed decreases while moving in the positive direction, the acceleration is negative. If the speed increases while moving in the negative direction, the acceleration is negative. If the speed decreases while moving in the negative direction, the acceleration is positive. The infinitesimal definition of the acceleration is $a=dv/dt$.

Now we move on to developing the equations of kinematics, basically there are five equation which one has to remember,

1. we almost always take acceleration to be constant, $a=$const for our purpose.
2. $v=u+at$, final velocity in terms of $u,a,t$
3. $s=((u+v)/2)t$, distance in terms of $u,v,t$
4.$s=ut+(1/2)at^{2}$, distance in terms of $u,a,t$
5.$v^{2}=u^{2}+2as$ final velocity interms of $u,a,s$

where, $u=$initial velocity, $a=$acceleration, $t=$time.,$s=$distance, $v=$final velocity, and $t=$time

Now if you're interested in the derivation, click here to watch a video

HERE's a quick exercise to solve: Is it possible that the car could have accelerated to 55mph within 268 meters if the car can only accelerate from 0 to 80 mph in 15 seconds?  (1mph->.447m/s)

soln: acceleration of the car is 2.38$m/s^{2}$
where as the required acceleration for (0to55mph) in 268m is 1.127$m/s^{2}$
Now since the acceleration required is less than it actually is, it is possible  that the car could accelerate to 55mph within 268 meters