A cricket ball of mass $145\; gm$ is moving with a velocity of $14\;m/s$ and is being hit by a bat, so that the ball is turned back with a velocity of $22\;m/s$. The force of blow acts on the ball for $0.015\; sec$. Find the average force exerted by the bat on the ball.

Given,

Mass of cricket ball $(m) = 145 \; gm = 0.145 \; kg$

Initial Velocity of ball $(u) = 14 \; m/s$

Final velocity of ball $(v) = 22\;m/s$

Time of impact $(t) = 0.015 \; sec$

Average force exerted by the bat on the ball $(F) = ?$

We have, $F \; = \; $$\frac{m(v- u)}{t}$

$ = $$\frac{0.145\;[22-(-14)]}{0.015}$           [∵ Being two velocities are opposite direction] 

$= 348\;N$

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Why is time repeated twice in the unit of acceleration?

In nature, we see that if something moves, it changes its location. It takes some time to complete that movement. So the change in location over a time is defined as speed (or, its rate of change). If the thing is moving in a particular direction then the speed is defined as velocity.

Mathematically, 

Velocity (v) = $\frac{distance (x)}{time (t)} $ 

Velocity is the rate (or speed) an object is moving from A to B over a measurable time.

It's not possible to maintain a constant speed for a very long time. At some point, the speed will increase (or, decrease) or change the direction of motion. All of these changes take place over a time, which are in the form of acceleration.

Mathematically,  

Acceleration (a) = $\frac{velocity (v)}{time (t)} = \frac{distance (x)}{time (t)\; * \;time (t)}$

Acceleration is the rate (or speed) at which an object is increasing or decreasing its velocity over a measurable time.

For example: Firstly, we notice that the object change its original position with the certain motion. In the first second, it moved at a speed of 1 meter per second. But after ten seconds, the object travels 2 meters per second. So in 10 seconds, it's speed has increased by 1 meter per second. This is called acceleration.


Finally, let's try to explain this concept in a scientific point of view:

Acceleration as doing two things at once. "We are still moving across a distance over a time, but we are also increasing how fast we are doing it." We are multi-tasking to arrive sooner. So, time is repeated twice in the unit of acceleration.

Hence, time appears twice: once to describe the rate at which position is changing (i.e. the speed) and once to describe the rate at which the speed is changing.

E1.3 Kinematics (Speed and Velocity):

1. Speed: Speed is a scalar quantity. This is a distance traveled by an object per unit time. Speed does not depend on the direction but on the total length in any direction. The unit of speed is $m/s$.

If $d$ is the distance traveled in time $t$, then speed $(S)$ = $\frac{distance\; (d)}{time \;(t)}$.

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(i) Average Speed: total distance traveled by the object in any direction in one second. It is defined as the rate of change of distance of the object from the fixed point. It's unit is $m/s$.

Let $S$ be the total distance traveled by the object in time interval $t_1$ to $t_2$. 

i.e. $S_{av}$ = $\frac{S}{t_2 - t_1}$ (∵ ${t_2 - t_1}$ = $ \Delta t$).

(ii) Instantaneous Speed: The speed of the object at a particular moment.It's unit is $m/s$. Let $\Delta S$ be the very small distance travelled by an object in a very short time interval $\Delta t$, approaches to zero.

i.e. Instantaneous Speed ($S$) = $\lim_{\Delta t\rightarrow 0}$ = $\frac{\Delta S}{\Delta t}$

2. Velocity: Velocity is a Vector quantity. This is a Displacement of the body per unit time. Velocity depends on the direction of motion. The unit of velocity is $m/s$. 

Let $D$ be the displacement traveled in time $t$, then velocity $(V)$ = $\frac{D}{t}$. [i.e. $\frac{Distance\; + \;Direction}{time}$]. 

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(i) Average Velocity: It is defined as the ratio of its total displacement to the total time interval in which the displacement occurs.

$S_{av}$ = $\frac{S_2 - S_1}{t_2 - t_1}$ = $\frac{\Delta S}{\Delta t}$.

(∵ $ \Delta S$ = total displacement and $ \Delta t$ = total time)

(ii) Instantaneous Velocity: The velocity of an object at any given instant of time at particular point of its path is called instantaneous velocity.

Instantaneous Velocity($S$) = $\lim_{\Delta t\rightarrow 0}$ = $\frac{\Delta S}{\Delta t}$

Uniform and Non-Uniform Velocity:

Uniform Velocity: If a body travels equal distances in equal intervals of time.

Non-Uniform Velocity: If an object does not travel equal distances in equal intervals of time or the direction of motion changes, the body is said to be in non-uniform (Variable) velocity.

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