V-T Graphs: Definition, Types and Sample Questions

V-T graphs in physics are a graphical representation of various  types of motions ; it gives a quick description of the motion undertaken by the body/object of study. It is easier to study, formulate and therefore, is used to give quicker and simpler information about the state of a body. Based on the relative idea of  rest and motion , the V-T graph deals with the velocity-time graph and provides details regarding the motion induced in the body in accordance with velocity, corresponding to time. 

Read Also: Speed Distance Time Formula

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Key Terms: Velocity-Time Graph, Tyres, Equations of motion, Special Cases, Acceleration, Velocity

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V-T Graph

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A vt graph is a  graphical representation of different types of motions; the body/object under study exhibits motions by falling under a VT (velocity-time) frame. It means that the graph will discuss the motion of the body by seeing the change in its velocity with respect to time. 

The y-axis represents the velocity of the object, x-axis represents the time and the slope of the graph is nothing but  acceleration . 

While there are three main equations of motion to represent, some other important parts remain too. Velocity-time graph represents the first equation of motion, v = u + at. 

The three equations of motion are:

v = u + at.

v² = u² + 2as. (represented by the position-time graph)

s = ut + ½at² (represented by velocity-position graph)

Also Read:

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Various Cases of V-T Graph

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In this section we will check all the possible cases and graphical representations that we can draw with the help of a velocity-time graph. 

When the object is at rest

When the object is at rest, it means there are two possibilities:

• Velocity, v = 0, or,
• Velocity, v = constant with respect to time. 

Thus, the velocity-time graph will be like this:

This showcases that the object remains at rest and has no external force applied to change its state of motion.

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When Object is in Motion

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Once the object has been set in motion by some external force, there are mainly three cases based on the  three equations of motions  that the velocity-time graph represents:

When Object Moves with Uniform Velocity

When the object moves with uniform velocity, it means that the acceleration of the body is zero. Thus, it object moves with a constant velocity throughout with slope, i.e. acceleration, a = 0.

So, from 

v = u + at

v = u, means initial velocity is equal to final velocity at all times. 

The velocity axis forms a straight line to coincide with the time axis.

When Object moves with Uniform Acceleration

Uniform change in acceleration  of the body means that the object changes its velocity at regular intervals of time. When object moves with uniform acceleration, there two further cases to study:

1. When the object has positive (or upward) slope uniformly

When the object moves with positive or upward slope constantly, it showcases positive uniform acceleration. Means, the  object accelerates  at regular intervals in the direction of motion. 

• When the object has negative (or downward) slope uniformly 

When the object moves with negative or downward slope constantly, it showcases negative uniform acceleration. Means, the object de-accelerates at regular intervals opposite to the direction of motion. 

• When object moves with changing acceleration

When acceleration of the body keeps changing, there are two more cases possible:

• When the object moves with positive (or increasing) acceleration

When the body moves with positive or increasing acceleration, it is generally the case when it moves forward. For example, a car driver peddling on the accelerator of the car to make it move ahead. 

The magnitude of the slope continues to rise as  acceleration increases . Also, initial velocity is often not zero in this case. Thus, by 

v = u + at

u = 0

So, v = at making acceleration a function of time. 

• When the object moves with negative (or decreasing) acceleration.

When the body moves with a negative acceleration, it is often termed as ‘retardation,’ means decreasing acceleration. It is generally the case when it starts to come to a halt. 

For example, a car driver applying brakes for the car to either slowly come to stop or slow down the speed. 

A changing acceleration may combine both accelerating and retarding forces to induce a change in the motion of the body by causing its velocity to fluctuate at irregular intervals of time. Thus, when a body moves with non-uniform acceleration, it is said to be in non-uniform motion. 

Also Read:

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Special Cases

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Unlike other graphs, this one has its coordinates in the other quadrant too. So, what kind of graph is it?

It is a decelerating graph with constant acceleration. As the graph scales lower and touches 0 at the time axis, it represents the object has been de-accelerating and has finally come to stop. For example, a car driver applying brakes on the car and it stops. 

So, v = 0 

But, when the object further moves into the other quadrant, going beyond the time axis, it showcases that the object has begun moving again with constant acceleration, in the backward direction. 

For example, a car driver reversing his car. 

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Representation of First Equation of Motion

As we know velocity-time graph represents the first equation of motion, let’s derive it both algebraically and graphically:

Algebraic method:

As, acceleration = change in velocity/time

a = Δv/t

a = (v-u)/t

Rearranging gives us, 

v = u + at.

Graphical method:

Consider a velocity-time graph like this:

• Velocity changes from A-B with respect to time, t (uniformly)
• Consider BC, v and OC, t
• Draw a perpendicular from B to OE and a parallel line is drawn from A to D. Thus, A perpendicular is drawn from B to OC.

BC = BD + DC

Hence, 

v = BD + DC

and 

v = BD + OA (since DC = OA)

Therefore, 

v = BD + u (since OA = u) (Eq 1)

Now,

Let a = slope of line AB 

[ { V(final) - V (initial) } ]/time = acceleration, a (this is the slope formula)]

a = BD/AD

Since

AD = AC = t,

BD = at (Eq 2)

Equation 1 + equation 2, we get: 

v = u + at

Also Read:  Difference Between Acceleration and Velocity

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Things To Remember

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Here are some of the important things to remember:-

• The velocity-time graph is a graphical representation of the motion exhibited by a body with respect to the change in velocity , acceleration of slope by corresponding to time. 
• When the object is in constant motion, the velocity-time graph is a straight line with no acceleration. 
• When the object does have acceleration, if it is constant, it either goes to positive or negative slope depending upon acceleration and deceleration exhibited respectively. 
• When the object has acceleration but it is non- uniform , it is either positive or negative depending upon the slope produced upon acceleration or deceleration respectively. 
• To  find the slope , perform { V(final) - V (initial) }/time, which will give acceleration. 

Also Read: Derivation of Equation of Motion

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