A ball is projected vertically upwards with a velocity v. It comes back to ground in time t. Which v-t graph shows the motion correctly?

1. Average speed:

It is the ratio of the total distance to the total time. 

2. Average Velocity:

It is the ratio of the total displacement to the total time. 

3. Instantaneous Velocity:

(i) It is the rate of displacement with respect to time

(ii) Mathematically expressed as, $\overrightarrow{v}=\underset{\Delta t\to 0}{\lim }\frac{\Delta x}{\Delta t}=\frac{dx}{dt}$

4. Average acceleration:

It is the ratio of total change in velocity to the total time taken.

5. Instantaneous Acceleration:

(i) It is the rate of change of velocity

(ii) Mathematically, it is expressed as, $a=\frac{\mathrm{d}v}{\mathrm{d}t}=\underset{\Delta t\to 0}{\lim }\frac{\Delta v}{\Delta t}$

6. Motion from a graph:

(i) Slope of distance-time graph gives speed

(ii) Slope of displacement-time graph gives velocity

(iii) Slope of velocity-time graph gives acceleration

(iv) Area under velocity-time graph gives displacement

(v) Area under speed-time graph gives distance

(vi) Area under acceleration time gives change in velocity.

7. Equations of Motion (for constant acceleration):

(i) v=u+at

(ii) $s=ut+\frac{1}{2}a{t}^2$

(iii) v2=u2+2as

(iv) s=u+v2t

(v) sn=u+a2(2n-1)

8. Relative motion:

(i) vAB (velocity of A with respect to B) =vA-vB

(ii) aAB (acceleration of A with respect to B) =aA-aB

9. Projectile motion:

(i) Time of flight T=2usinθg

(ii) Horizontal range R=u2sin2θg

(iii) Maximum height H=u2sin2θ2g

(iv) Trajectory equation (equation of path): y=xtanθ-gx22u2cos2θ=xtanθ1-xR

(v) Velocity after time $t$, $v=u\cos \theta \hat{i}+\left(u\sin \theta -gt\right)\hat{j}$

10. Relative motion:

(i) vAB (velocity of A with respect to B) =vA-vB

(ii) aAB (acceleration of A with respect to B) =aA-aB

11. Crossing river:

(i) A man in a river always moves in the direction of resultant of velocity of man and velocity of river.

(ii) Velocity with which man moves along the river Vmx=Vr-Vm,Rsin⁡θ

(iii) Velocity with which man moves perpendicular to the river Vmy=Vm,Rcos⁡θ

(iv) Time take to cross the river, t=WVm,Rcos⁡θ

(v) To get minimum time $\frac{\mathrm{d}\mathrm{t}}{\mathrm{d}\mathrm{\theta }}=0\Rightarrow \mathrm{\theta }=0$

(vi) Drift in this case, $d=\frac{{\mathrm{V}}_{\mathrm{r}}\mathrm{W}}{{\mathrm{V}}_{\mathrm{m},\mathrm{R}}}$

(vii) For shortest path, drift must be zero in this case, and it happens when ⇒sin⁡θ=VrVm,R