Graphical Treatment of Uniformly Accelerated Motion

Acceleration of a body is defined as the rate of change of velocity with respect to time.

Uniformly accelerated motion –

The motion in which the acceleration of a particle is constant throughout the motion.

For example, if we throw a ball vertically upwards, then the acceleration due to the gravity is constant from the beginning till the end.

For this particular type of motion, we can define three types of graphs –

1. Acceleration-time graph –

As the name suggests, the acceleration will not change with time or it will remain constant. So, the graph will look like:

On the x-axis, we take time; 

And on the y-axis, we take acceleration.

Depending upon the direction of acceleration, the graph could lie in either the first quadrant or third quadrant.

In the given example, acceleration due to gravity is -9.8 m/ s

2. Velocity-time graph –

When we throw a ball, the velocity of the ball decreases first, becomes zero at the topmost point and then starts to increase in the opposite direction. To show this on the graph, let us suppose the initial velocity with which we are throwing the ball is 5m/sec. When we catch the ball, it will have the same velocity, but in the opposite direction. So, the graph will look like this –

The nature of the graph will be a straight line because we have Newton’s equation of motion. 

v=u+at

s=ut+1/2at2

v2=u2+2as

From the first equation, we can see that there is a linear relationship between velocity and time.

3. Displacement time graph – From the 3rd equation, we can see there is a parabolic relationship between displacement and time. First, the displacement will increase, become maximum at the top and then starts to decrease and become zero when we catch the ball.

The above graph also verifies that at the topmost point, the velocity should be zero and if we draw a tangent at the highest point, we see the slope of the tangent is zero at that point. This indicates that the velocity at the highest point must be zero.

Projectile Motion

When we throw a ball not vertically upward, but at some angle θ, we will get a parabolic path of the ball. This time, the ball will be doing two-dimensional motion. 

The velocity has two components, one in the x-direction and another in the y-direction.

But the value of acceleration due to gravity will be constant. The direction of ‘g’ is perpendicular to the x-axis a, so it will not affect the magnitude of the velocity along the x-axis.

Range – it is the maximum displacement along the x-direction done by the projectile.

Maximum height – It is the maximum displacement along the y-direction done by the projectile.

Time of flight – It is the time taken by the projectile to complete the given path.

Time of the fight, T=2usinθg

Maximum height attained, Hmax = u2 2g

Horizontal range, R=u22θ g

Where u is the initial velocity with which the projectile was thrown.

‘g’ is the acceleration due to gravity

Inclined projectile motion:

When we throw a projectile at an inclined plane, then this motion is again an example of uniformly accelerated motion.

The difference between the general projectile and the inclined projectile is that in the inclined, the velocity along the chosen x-direction is affected by the acceleration due to gravity.

The motion of a charged particle in an electric field 

 If a charged particle having charge q and mass m is at P point under an applied external electric field E, then it will feel a force F 

                     F = qe

The charge of the particle is constant, E is fixed and if we divide this F by m, then we will get the acceleration, which would be a constant value. Then, this situation also falls into the category of uniformly accelerated motion.

Conclusion

Uniformly accelerated motions are versatile kinds of motion. What we see around us, such as a person throwing a ball straight up in the sky, fountains, cyclotrons to produce high energy particles in laboratories, the motion of a bird in the sky, all are examples of motion under uniform acceleration (under some constraints). Although we also have a large platform where the uniform acceleration motion is not allowed, we can easily realise why we study this topic in the course of physics.