Kinematics-Motion In A Straight Line

Let us begin understanding the acceleration motion in a straight line by answering, “What is Acceleration?”

 We all know the speed at which cars accelerate and go up when we press down to the pedal. The change in the speed of a particle relative to its time is referred to as its acceleration. If the speed of the particle is changing at a constant time, it is referred to as an acceleration constant. Because we’re using metres and seconds as the basic units, we’ll measure acceleration in metres per second (m/s2). It is also often written as ms-2.

 For instance, if you can see fluctuation in the velocity of a particle that is moving through a uniform line in the same time between 2 and up to 5 m/s within a single second, the constant acceleration of that particle is 3 milliseconds.

 Decreasing Velocity

Suppose the velocity of a particle that is moving along a straight line changes in a uniform manner (at the same time) between 5m/s up to 2 m/s within a single second, and its constant acceleration is -3 m/s2.

If particles have a starting velocity of 6 m/s with an acceleration constant of -2 m/s2 

When t = 1, then the speed of the particle will be 4 m/s 

When t = 2, the particle’s velocity is 2 m/s. 

When t = 3, the particle’s velocity is 0 m/s. 

When t = 4, the particle’s velocity is -2 m/s . 

When t = 10, the particle’s velocity is -14 m/s. 

 The velocity for the particle will be 6 – 2 m/s following two seconds. 

For the first three seconds, the speed of the particle decreases (the particles are slowing). In three seconds, the particle is briefly still. After three seconds, the velocity declines—however, the speed increases (the particle moves faster and more quickly). 

If the rate of variation in the velocity is constant and the acceleration is constant, then the constant is calculated by Acceleration change/change in time.

 Acceleration Formula

Throughout this section, we have been considering motion in a straight line with constant acceleration. This is a common scenario, such as anybody that is under the pressure of gravity moves at a constant speed. 

There are five common acceleration formulas that describe moving in straight lines that have constant acceleration.

 These formulas are presented as a function of the initial velocity u, the last velocity v, the distance of x along with the acceleration a, and the duration of time t. Of course, there must be the use of a consistent system of units to be utilised. The assumption is that motion starts at the point t = 0 and that the position at which it began is considered to be the starting point of the motion, i.e., x(0) = 0.

The five equations of motion 

1. v = u + at

2.  x = 12(u+v)t

3. x = ut + 1/2at2

4. v2 = u2 + 2ax

5. x = vt – 1/2at2

The five formulas involve five variables: such as a, v, x, t, and u. When the numbers of three factors are known, the other values are obtainable by using two equations.

 Deriving the constant-acceleration formulas 

The first equation of motion 

Since the acceleration is constant, we have a = (v -u)/t . 

Second equation for motion

The second equation is x = (u + v)t/2

It is said that displacement can be calculated by adding final and initial velocity times the time during the movement. 

More simply: Displacement = Average velocity x Time taken.

The 3rd equation of motion. 

Substituting for v from the initial equation into the second yields

x = (u+v)t/2

= (u + u + at)t/2

= 2ut + at2/2

= ut + 1/2 at2

The fourth motion equation. 

From one equation we can find the equation t = (v-u)/a . Incorporating this equation into the second equation, we get

x = (u + v)t/2

= (u+v)(v-u)/2a

=v2 – u2/2a

Rearranging the equation to make v2, the subject creates the fourth equation: V2 =  u2 + 2ax.

The fifth motion equation: 

From our first formula, we can find the equation v = u + a. Utilising this equation we can get

x = (u + v)t/2

= (v – at + v)t/2

= 2vt – at2 /2

= vt – ½ at2 

 Conclusion

The rate of change of the velocity of a particle with respect to time is called its acceleration. Since the basic units that we are using for the calculation are metres and seconds, the acceleration will also be calculated in metres per second. If we assume that the rate of change of velocity (acceleration) is a constant, then the constant acceleration is given by,

Acceleration = change in velocity/change in time. 

There are five frequently used formulas for motion in a straight line with constant acceleration. The equations are given in terms of the initial velocity u, the final velocity v, the displacement x, the acceleration a, and the time elapsed t.