Motion

Rate of motion is called speed. When this rate of motion i.e. speed is represented along with direction, then it is called VELOCITY.

Thus, Velocity is the representation of rate of motion with direction.

In other words, the quantity that specifies both rate of motion and direction is called VELOCITY.

Using velocity the rate of motion of an object can be represented more comprehensive.

Velocity has both magnitude and direction.

Average Velocity

When an object is moving along a straight line at a variable speed, we can express the magnitude of its rate of motion in terms of average velocity.

In case the velocity of the object is changing at a uniform rate, then average velocity is given by the arithmetic mean of initial and final velocity for a given period of time. That is

Average velocity `=(text{initial velocity + final velocity})/2`

Initial velocity is generally represented by letter `u`.

Final velocity is generally represented by the letter `v`

And Average Velocity is generally represented by `v_(av)`

Thus, Mathematically

`v_(av) = (u+v)/2`

Velocity `=(text{Displacement})/(text{Total time taken})`

SI unit of average velocity:

Speed and velocity have same SI units, that is, `m//s` or `m\ s^(-1)`

Rate of Change of Velocity

Acceleration

Rate of change of velocity is called Acceleration.s

Rate of change of velocity, i.e. Acceleration during Uniform Motion

During uniform motion of an object along a straight line, the velocity remains constant with time. In this case, the change in velocity of the object for any time interval is zero.

This, means in the case of uniform motion, the rate of change of velocity, i.e. acceleration is equal to zero.

Rate of Change of Velocity, i.e. Acceleration during Non-Uniform motion

In the case of non-uniform motion, velocity varies with time. Thus, velocity has different values at different instants and at different points of path. Thus, in this case the change in velocity of the object during any time interval is not zero.

Thus, acceleration (`a`) is the change in velocity of an object per unit time.

Thus,

Acceleration (`a`) = `(text{change in velocity})/(text{time taken})`

Change in velocity = final velocity (`v`)– initial velocity (`u`)

Or, Change in velocity = `v-u`

Thus, Acceleration (`a`) `=(v-u)/t`

Where, `a` = Acceleration (rate of change of velocity)

`u` = Initial velocity

And `v` = Final velocity

And, `t` = Time

This kind of motion is called accelerated motion.

Positive and Negative Acceleration

The SI unit of Acceleration

The SI unit of Acceleration (`a`) is m/s/s or `m\ s^(-2)`

Uniform and Non-uniform acceleration

If an object travels in a straight line and its velocity increases or decreases by equal amounts in equal interval of time, then the acceleration of the object is said to be uniform or uniform acceleration.

The motion of a freely falling body is an example of uniformly accelerated motion.

On the other hand if velocity of an object changes at a non-uniform rate, thus the acceleration of the object is called non-uniform acceleration.

Graphical Representation of Motion

Motion can be represented using graph. The line graphs show dependence of one physical quantity, such as distance or velocity, on other quantity, such as time.

Distance – Time Graphs

The change in the position of an object with time can be represented on the distance-time graph. Distance–Time Graph of an object moving with uniform speed.

In the graph time is taken along the `x`-axis and distance is taken along the `y`-axis.

When an object travels with uniform speed, it covers equal distance in equal interval of time. This shows that the distance travelled by an object is directly proportional to time taken.

Thus, for uniform speed, a graph of distance travelled against time is a straight line.

In the given graph the portion OB shows that the distance in increasing at uniform rate.

Determining the speed of an object Using Time–Ditance graph

To determine the speed of the object, Let consider a small portion AB on the distance-time graph.

A line parallel to the `x`-axis is drawn from point A and another line parallel to the `y`-axis is drawn from the point B.

These two lines meet at point C forming a triangle ABC.

Now, according to distance-time graph

AC denotes the time interval = `t_2-t_1`

And BC denotes the distance = `s_2-s_1`

Thus, according to graph, it is clear that

From point A to B, object covers a distance of `(s_2-s_1)` in time `(t_2-t_1)`.

That is distance `=s_2-s_1`

And time `=t_2-t_1`

Thus, speed `(v)` of the object `=(text{distance})/(text{time})`

`=> v = (s_2_s_1)/(t_2-t_1)`

Distance–Time Graph for Accelerated Motion

When a distance time graph is plotted for an object moving with accelerated motion, i.e. with non-uniform speed, line of graph is not a straight line.

Let a car travels with non-uniform speed, distance covered by car at a regular time interval is given in the table.

When a graph is plotted against the distance and time given in the table, the nature of the graph shows non-linear variation of the distance travelled by the car with time.

Velocity–Time Graph

When object is moving with uniform velocity, the velocity-time graph does not change with time and it will be a straight line parallel to `x`-axis. In this graph velocity has been taken along `y`-axis and time has been taken along `x`-axis.

In the given graph the object is moving with a constant velocity 40 km/h.

Knowing the distance covered by object using velocity-time graph

To know the distance covered by the object between time `t_1` and `t_2` using the given graph, perpendiculars from the points corresponding to the time `t_1` and `t_2` are drawn.

The velocity of 40km/h is represented by the height AC or BD and the time interval `t_2-t_1` is represented by the length AB.

Since, the product of velocity and time gives the displacement of an object moving with uniform velocity. Thus, the area enclosed by velocity time graph and the time axis will be equal to the magnitude of displacement.

So, the distance (`s`) moved by the object in given time interval, `t_2-t_1` can be expressed as,

`s = AC xx CD`

`=[(40\ km\ h^(-1)) xx (t_2-t_1)h]`

`= 40(t_2-t_1)\ km`

= Area of the rectangle ABCD (shaded in graph).

Thus, using velocity time graph distance covered by object moving with uniform velocity can be calculated.

Velocity–Time Graph for Uniformly Accelerated Motion

A velocity time graph of an object moving with uniformly accelerated motion is also a straight line, but nature of this graph is different.

Let an object is moving with uniformly accelerated motion, of which velocity at a regular interval of time is given in the table.

Graph of the object of which velocity and time is given in the table is as follows:

In this graph velocity is taken along `y`-axis and time is taken along `x`-axis. The nature of the graph clearly shows that velocity of the object is changing by equal amounts in equal interval of time.

Thus, for all uniformly accelerated motion, the velocity-time graph is a straight line.

Determining the distance covered by an object using velocity-time graph

The area under the velocity time graph gives the distance (magnitude of displacement) moved by an object in a given interval of time. This means that distance (magnitude of displacement) covered by an object can be determined using velocity time graph.

Thus, distance covered by the object would be represented by the area ABCD in the graph.

Since, the magnitude of the velocity of the object is changing due to acceleration, the distance `s` travelled by the object will be given by the area ABCDE under the velocity time graph as given above.

That is, Distance `s` = area of ABCDE

= area of the rectangle ABCD + area of the triangle ADE

`=AB xx BC + 1/2 (AD xx DE)`

Other shapes of velocity-time graph

If an object is moving with uniform negatively accelerated motion, the velocity-time graph is a straight line, but downward.

If an object is moving with non uniform variation in velocity with time, the velocity time graph is a zig jag type.

Thus, velocity time graph can be of any shape.