# Question #2e2cc

##### 1 Answer

#### Answer:

See below.

#### Explanation:

**Kinematics** is a branch of *mechanics* used to describe motion—without regard to what caused that motion (forces). **Kinematic equations** are used to mathematically describe the motion of an object (e.g. displacement, velocity, acceleration).

Consider an object whose **acceleration** **constant** during the **time interval**

At the beginning of this interval,

Say we want to predict the object's final position

The object's **velocity is changing because the object is accelerating** . We can find the object's velocity

By definition:

#a_s=(Deltav_s)/(Deltat)=(v_(ts)-v_(is))/(Deltat)#

which is easily rearranged to give

#color(blue)(v_(fs)=v_(is)+a_sDeltat)#

The velocity-versus time graph is a straight line that starts at

The object's final position is

#s_f=s_i+"area under the curve " v_s " between " t_i " and " t_f#

The **area under the curve** can be subdivided into a rectangle of area

Adding these gives:

#color(blue)(s_f=s_i+v_(is)Deltat+1/2a(Deltat)^2)# where

#Deltat=t_f-t_i# is the elapsed time

The final kinematic equation for constant acceleration is first found by using our first equation (in blue) to write:

#Deltat=(v_(fs)-v_(is))/a_x#

Substituting that into our second (blue) equation, we get:

#s_f=s_i+v_(is)((v_(fs)-v_(is))/a_x)+1/2a_s((v_(fs)-v_(is))/a_x)^2#

With a bit of algebra, this is rearranged to read

#color(blue)((v_(fs))^2=(v_(is))^2+2a_sDeltas)# where

#Deltas=s_f-s_i# is thedisplacement

Hope that helps!