Constant acceleration graph: what, how, examples - (2023)

The movement of an object can be described with the help of a graphical representation. The graph of position time, velocity time, and acceleration time is commonly used to describe motion.

It is said that a body only accelerates when the speed of the body changes. Ochange in speedover time can be plotted on a graph to describe movement. Acceleration can also be plotted in a similar way. In this post, we will know how to draw constant acceleration diagram in different aspects.

What is a constant acceleration graph?

Constant acceleration means when the rate of change of a body does not change much in a given time interval. In general, the acceleration of a body can be represented by the velocity-time diagram. The slope of the v-t diagram indicates the acceleration.

Motion of a body in a straight line with uniform acceleration is commonly referred to as constant acceleration. A non-zero uniformaccelerationa body moving in the plane can be graphed; This graph is known as a constant acceleration graph.

If the velocity of the moving body is constant, then we get zero acceleration.

Suppose a car is moving at a certain speed and increases its speed by 25 m/s in the next 5 minutes within the 5 minute time interval. In this case, the car needs to increase its speed by 25 m/s; In this case, the acceleration of the car is called a constant.

If we graph the increase in velocity over time, we get the acceleration from the slope of the graph, which represents constant acceleration.

How do you know if the acceleration is constant on a graph?

We've already looked at what a graph of constant acceleration means. Can you see from the diagram that the acceleration is constant?

The gradient of the velocity-time diagram indicates the acceleration. By differentiating the slope obtained, we can tell from the graph whether the acceleration is constant or not.

From the graph above, we can calculate the slope as

Slope = AB/BC

In the graph AB = v and BC = t; therefore, the slope can be written as

slope = v/t

If we now derive the given slope equation, we get;

du/dt = 0

The derivative of the slope is zero, which means that the acceleration is constant. If the derivative of the slope has a certain value, the acceleration is not constant.

Now look at the position versus time graph; the slope of local time gives the speed. If the velocity is constant on the position-time diagram, the acceleration on the diagram is also constant.

The slope of the x-t graph can be written as

declive=y/x= x/t

Now differentiate the slope obtained, we get the velocity as

dx/dt=v

Differentiating the above equation again, we get

D²x/dt²=dv/dt

But the first order derivative of the velocity gives the acceleration; therefore, the equation can be rewritten as

D²x/dt²=dv/dt=a

This implies that, on a position versus time graph, if the second-order time derivative of distance is constant, then acceleration is constant. Or, in other words, if the derivative of the slope of the position versus time graph is constant, then the acceleration on the x-t graph is constant.

When is acceleration constant on a graph?

Acceleration is constant on the graph when the speed of the moving object is increasing or decreasing at a constant rate.

If you draw the graph of velocity versus time and the curve you get from the graph is a straight line, the slope is constant throughout the graph. The acceleration is therefore constant on the diagram.

If you plot the position versus time graph, the slope of the graph indicates velocity. SinceChange in speed gives acceleration, then the x-t diagram gives the constant acceleration as shown below.

If the curve obtained by drawing an x-t graph is a parabola, as shown in the figure above, then from the graph we can say the following:

• Between points A and B, x has a negative value. The object starts moving towards the origin, causing the object's velocity to become positive. The slope of the object is positive and therefore the acceleration is positive and it has a constant acceleration.
• At point B, the object hits the origin and x takes on a positive value. In this position, the object gains maximum speed and the speed increases proportionally. Therefore, the acceleration is constant.
• Between points B and C, the value of x increases and the object starts moving to the right of the origin. Thus, the object's velocity decreases at a constant rate and the slope of the graph also decreases, although the acceleration remains constant.

Therefore, from the two examples in the graph above, it is clear that the acceleration is constant when the object's motion follows a straight-line path or a parabolic curve with a change in velocity.

How to draw constant acceleration?

There are several methods for plotting the constant acceleration graph. Three important ways to represent constant acceleration are:

• Position versus time diagram
• Speed ​​x time diagram
• Acceleration time diagram

Now let's discuss how to plot a constant acceleration using these three graphs in sequence.

Position versus time diagram

The position versus time diagram is usually plotted to describe motion and its slope is used to calculate the velocity of a particle.

Since the second-order time derivative ofDistance indicates acceleration, we can plot the constant acceleration graph using the position versus time graph. The start and end position where the object was within the given time interval is used to plot a constant acceleration.

The x-t graph showing the constant acceleration is shown below.

In the graph above, position versus time gives the parabolic curve, increasing velocity and positive acceleration. From the equation of motion, the distance traveled by the body is given by the equation

In the above equation, x is the final position and x₀is the starting position of the body.

Using the expression above, the constant acceleration can be plotted on the graph using the change in magnitude of the velocity.

Speed ​​x time diagram

An object is accelerated uniformlystraight motioncan be plotted using velocity on the vertical axis and time on the horizontal axis.If the velocity increases linearly with successive time intervals, then the acceleration is constant.

The kinematic equation of motion is used for illustration.constant acceleration on v-t diagram.

v = v₀+ em

Looking at the graph above, theinitial speed v0Eterminal velocity vwith the given time t form the right triangle that helps to describe the constant acceleration in the diagram.

from the triangle

From the above expression, we can express the change in velocity as;

The above equation gives the change in velocity, whose slope gives the acceleration. The acceleration in the diagram is constant.

Acceleration time diagram

The graph of acceleration time for constant acceleration is a straight line. Acceleration does not change with time.The area covered by the a-t diagram gives the change in velocity.The constant acceleration acceleration time graph is given below.

Acceleration does not change with time to give constant acceleration.

Some examples of constant acceleration graphs

Example 1) When a person drives the car on a busy road, at first he drives the car with a smooth increase in speed. When he reaches the city limit speed, his speed becomes constant. If he suddenly encounters speed bumps on the road, then his speed will decrease. The v-t diagram for this situation is given below.

From the graph above, we can analyze some facts, for example

• The car starts moving from point A; this point would be the initial speed of the car.
• From point A to B, the velocity increases smoothly over time, and the slope also increases, so the acceleration between A and B is constant.
• Between points B to C the velocity is kept constant and therefore there is no acceleration.
• From point C to D, the speed of the car decreases and the slope decreases, so we get a constantly decreasing acceleration.

Example 2) The movement of an object is shown in the graph below,

By analyzing the graph above, we can understand things like:

• At point "0", the starting position from which the object starts moving.
• Between points 0 and A we can observe the increase in velocity expressed by the increase in incline when the incline increases steadily with time; the acceleration is constant.
• Between points A and B, the slope decreases, which means the speed decreases. However, the decrease in velocity is constant to observe constant acceleration.
• Between points B and c the slope increases and so we can observe the variation in velocity; therefore, the acceleration between B and C is constant. Between points C and D, the slope is constant, which implies that the velocity of the object is constant. Therefore, there is no acceleration between C and D.