A player throws a ball upwards with an initial speed of $29.4 \mathrm{m} {\mathrm{s}}^{-1}$.
(a) What is the direction of acceleration during the upward motion of the ball?
(b) What are the velocity and acceleration of the ball at the highest point of its motion?
(c) Choose the $x=0 \mathrm{m}$ and $t=0 \mathrm{s}$ to be the location and time of the ball at it’s highest point, vertically downward direction to be the positive direction of the x-axis, and give the signs of position, velocity and acceleration of the ball during its upward, and downward motion.
(d) To what height does the ball rise and after how long does the ball return to the player’s hands? (Take $g=9.8 \mathrm{m}{\mathrm{s}}^{-2}$ and neglect air resistance).

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, $\overrightarrow{v}= \underset{\text{Δ}\text{t}\to 0}{\lim }\frac{\text{Δ}\overrightarrow{x}}{\text{Δ}\text{t}}= \frac{\mathrm{d}\overrightarrow{x}}{\mathrm{d}t}$

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, $\overrightarrow{a}= \underset{\text{Δ}\text{t}\to 0}{\lim }\frac{\text{Δ}\overrightarrow{v}}{\text{Δ}\text{t}}= \frac{\mathrm{d}\overrightarrow{v}}{\mathrm{d}t}$

6. Motion from a graph:

(i) The slope of distance-time graph gives speed.

(ii) The slope of displacement-time graph gives velocity.

(iii) The slope of velocity-time graph gives acceleration.

(iv) The area under velocity-time graph gives displacement.

(v) The area under speed-time graph gives distance.

(vi) The 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) $v^2= u^2+2as$

(iv) $s= \frac{\left(u+v\right)}{2}t$

(v) $s_n= u+\frac{a}{2}(2n-1)$

8. Relative motion:

(i) ${\overrightarrow{v}}_{AB}$ (velocity of A with respect to B) $= {\overrightarrow{v}}_A-{\overrightarrow{v}}_B$

(ii) ${\overrightarrow{a}}_{AB}$ (acceleration of A with respect to B) $= {\overrightarrow{a}}_A-{\overrightarrow{a}}_B$