Graph -

With the help of graphs we visualise the variation of position (x), velocity (v), and acceleration (a) of a moving particle with time. Plotting time (t) on X-axis and x, v, a on Y-axis we get three useful graphs: (i) x-t graphs (ii) v-t graphs (iii) a-t graphs

Position – Time graph:- (s-t graph)

Generally, the position is indicated on the y-axis, and time is indicated on the x-axis.

We know that $\fn_cm \left [v= \frac{\mathrm{d}s }{\mathrm{d} t}\right ]$

i.e. slop of the tangent (tanθ with the +ve side of the x-axis) at any point on the s-t graph gives the instantaneous velocity at that instant. In the graph, more slope means more velocity.

Velocity – Time graph:- (v-t graph)

Generally, the velocity is indicated on the y-axis, and time is indicated on the x-axis.

We know that $\fn_cm \left [a= \frac{\mathrm{d}v }{\mathrm{d} t}\right ]$

i.e. slop of the tangent (tanθ with the +ve side of the x-axis) at any point on the v-t graph gives the instantaneous acceleration at that instant. In the graph, more slope means more acceleration

Also, we know that $\fn_cm v=\frac{ds}{dt}\;\;\therefore \;\;\int ds=\int vdt$

$\fn_cm \therefore \left [ s=\int_{t_1}^{t_2} vdt\right ]$

i.e. in the v-t graph area between $\fn_cm t_1$ and $\fn_cm t_2$ under the time axis gives the displacement between time $\fn_cm t_1$ and $\fn_cm t_2$

Acceleration – Time graph:- (a-t graph)

We know that $\fn_cm \left [a= \frac{\mathrm{d}v }{\mathrm{d} t}\right ]$ $\fn_cm \therefore \left [ v=\int_{t_1}^{t_2}adt \right ]$

i.e. area enclosed between acceleration time graph under time axis gives the change in velocity between time $\fn_cm t_1$ and $\fn_cm t_2$

NOTE:-

1. For uniform motion

2. Uniformly accelerated motion

3. Uniformly decelerated motion

4. Graph not possible

Questions related to graph

1. Find the distance travelled in 10 sec. ( 50 m)

2. From given graph

find velocity at t=1 sec, 4 sec, 7 sec, 9 sec, 12 sec, 15 sec and find average velocity between the given interval t=o to 2 sec, t=0 to 4 sec, t=0 to 8 sec and t=0 to 16 sec. ( 5 m/s,0,-5 m/s, -5 m/s, 0, 5 m/s, 5 m/s, 2.5 m/s, -5/3 m/s 0)

3. The s-t graph of a particle is given.

(a) At which point A,B,C and D is the velocity +ve
(b) At which point is the velocity -ve?
(c) At which point is the velocity 0?
(d) At which point acceleration is +ve?
(e) At which point acceleration is -ve.
(f) At which point a is zero.
(g)At which point, state whether the speed is increasing, decreasing or not changing.

4. The x-t graph of a particle is given

(a) Find the average velocity in the time interval 0 to 2sec, 2 to 4 sec and 4 to 7 sec (5 m/s, -2.5 m/s, -3.3 m/s)
(b) find velocity in each interval.

5. The position time graph of two children A and B returning from their school to their homes P and Q respectively.

Choose the correct option in the bracket below.

(a) (A/B) lives closer to school than (B/A)
(b) (A/B) start from the school earlier than (B/A)
(c) (A/B) walks faster than (B/A)
(d) A and B reach home at the (same / different) time.
(e) A and B overtake on the road (once/ twice)

(6) In the given s-t graph of two particles of A is B

(a) how much ahead of A when the motion starts
(b) What is the speed of A
(c) When and where will A catch B
(d) What is the difference between the speed of A and B

7. Describe the motion shown by the following (v-t) graph.

8. Find the acceleration at each interval of time

9. As soon as a car just starts from rest in a certain direction, a scooter moving with a uniform speed overtake the car.

Find

(a) The difference between the distances travelled by the car and the scooter in 15 sec.
(b) The distance of car and the scooter from the starting point at that instant when car catches scooter.

10.

Find

(a) Average velocity in total time
(b) Average speed in total time
(c) Draw (a-t) graph

11. The three particle A, B and C start at the same time , and its x-t graph is given below

(a) Which car has the highest speed and which has the lowest.
(b) Are the three cars ever at the same point on the road?
(c) When C passes A, where is B?
(d) What is the time interval during car A travel between the time it passed car B and C?
(e) What is the relative velocity of car C w.r.t car A?
(f) What is the relative velocity of car B w.r.t car C

(12) The x-t graph of two particles are given

(a) At what time(s), if any do A and B have the same position?
(b) At what time(s) if any do A and B have the same velocity? what is the velocity of car B at this time?

(13). Find the average acceleration in the 1st 20 sec.

(14). At t=0, the particle starts from rest and moves along st. line where (a-t) graph is shown

(a) Draw (v-t) graph
(b) From (v-t) graph find (a) max velocity (b) distance and displacement from 2 to 6 sec.

(15) Draw (v-t) and (a-t) graph

(16).A particle moves along x-axis with an initial speed 5 m/s. (a-t) graph is given .

(a) Find the velocity of the particle at t=4 sec.
(b) Find the time when the particle start moving along -ve x- direction.

(hint:- -ve area tells us that change in velocity is along -ve x-direction)

(17)

Find the signs of acceleration and velocity at points 1,2,3 and 4. Also comment about speed (increasing or decreasing)

(18) From the given (v-t) graph

Find (a) Average acceleration betwwen t=0 to 2 sec, t=0 to 4 sec and t=2 to 5 sec.
(b) Acceleration at t=1, 3 and 5 sec
(c) Displacement in 12 sec
(d) Distance in 12 sec
(e) Average velocity and average speed in 12 sec.

19. if u= 3m/s. find the velocity at t=4 sec from given (a-t) graph.

20. Convert the following (v-t) graph into (a-t) graph and (s-t) graph

21. Draw a graph of motion of a body which is dropped from some height. take initial position is origin and downward direction as +ve.

22. Draw the graph (s-t) ,(v-t) and (a-t) graph for the following situations.