If the displacement of a particle, x be a function of time t,

 

i.e. x = f (t), then, the velocity of the paricle, v = dx/dt = f ' (t)

 

& acceleration of the particle, a = dv/dt = d²x/dt² = f '' (t)

 

In the graphical representation of motion of a particle, the following points must be remembered :

 

1. In the displacement-time graph, the slope of the graph at any point gives the velocity of the particle at that instant of time.

2. In the velocity-time graph, the slope of the graph at any point gives the acceleration of the particle at that intant of time. 

3. In the velocity-time graph, the area under the curve gives the total displacement of the particle in the given time interval, since

_[b]^[a] v dt = x ,where, x is the displacement of the particle in the time interval

(a-b)

 

Now some common questions which you will come across.

 

1. When x-t graph is a straight line ( x-t graph can never be perpendicular to the time axis, as it means that the particle is at diff. positions at the same instant of time which is absurd ), then v = dx/dt is constant i.e v-t graph is a st. line parallel to the time axis, and a = dv/dt = 0

2. When x-t graph is a parabola of the form x=kt², then v = 2kt i.e v-t graph is a straight line , not parallel to time axis,  & a = 2k, i.e a-t graph is a st. line parallel to the time axis

3. When the v-t graph is a parabola of the form v = mt², then a=2mt, & a-t graph is a straight line passing through the origin.

 

Since you are in class 9, I believe, you don't know 'slope'.

'Slope' at a point means the tan ( of the angle that the tangent to the curve at that point makes with the positive X-axis )

I think you may not know about integrals much, in calculating area, mainly simple areas, formulas should be used. Like in case of a

square : area = l² sq. units , l = length of each side

rectangle : area = lb sq. units l = length b = breadth

triangle : area = 1/2 h b sq. units h = altitude b = base

parallelogram : area = b h sq. units  b = base , h = height

 

Well, the above are the basics of graphical representation of motion of a particle. I've shown how to go about this type of problems, so hope I am of some help to you.

 

Cheers !!!