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Let's examine the 3 kinematics formulae to summarize some strategies in playing tennis!!

Introduction

Tennis is an example of projectile launched at an angle above ground level. Despite being slightly different from one that’s launched at ground level, the same kinematics formulae are applied.

They are: -Vf=Vi+gt

-Vf*2=Vi*2+2gd

-d=Vit+1/2gt*2

Here is a diagram illustrating a tennis ball’s trajectory:

Therefore, the harder you hit, the lower you make the ball go, and a shorter time of flight are the KEY to winning tennis.

-The tennis ball’s vertical velocity at the maximum height is 0 m/s because at the top, it travels only horizontally for a specific period of time, so there is no Vy value.

-When the tennis ball falls down to the height that’s even with launch level, the magnitude of its vertical velocity is the same as that of its initial vertical velocity, but the direction is the opposite. This is because Vf*2=Vi*2-2gd, and d is 0 m. (THAT’S WHY THE STRENGTH YOU HIT THE BALL WITH=(well almost)THE POWER OF THE BALL WHEN IT HITS THE GROUND AND BOUNCES TO THE OPPONENT!!) But the very final impact velocity is going to be greater than the even level Vf because the ball is travelling further down towards the ground, accelerating due to gravity.

X- & Y-Components:

When a tennis ball is released at an angle with a given speed its trajectory should be analyzed in two parts: vertical(y) & horizontal(x) components, like the following shows:

-Horizontal motion:

-Vx=Vcostheta

-ax=0 m/s*2

-dx=Vxit (because according to dx=Vxit+1/2axt*2, ax=0)

-R=dx=Vxit

-Vxi=Vxf

-Vertical motion:

-Vy=Vsintheta

-ay=-9.8 m/s*2

-dy=Vyi+1/2at*2

-Vy=0 m/s at top

-Vyf=-Vyi at launch level

How are the principles of projectile motion applied in tennis?

1) Vf=Vi+gt

Because gravity is negative, let’s rewrite the equation this way: Vf=Vi-gt. In order to increase the final velocity of a tennis ball, the initial velocity must increase and as a result of an increasing initial velocity, the ball will take less time in its total flight, thus “gt” will decrease. Hence, when you subtract a deducted number from an enlarged number, the difference is going to INCREASE!

The greater the velocity/speed you hit the ball with, the harder the ball is going to hit on the opponent’s racket, thus increasing the level of difficulty.

2) Vf*2=Vi*2+2gd

Rearrange the formula as Vf*2=Vi*2-2gd (again because “g” is “-”). Besides greater initial velocity, a lesser displacement/height is also crucial to winning tennis (we can see from the equation). That’s why tennis ball players try to keep the ball as low as possible and in international matches, many rallies of the ball JUST skimmed over the net!

The lower the height you hit your ball over the net, the greater velocity it will gain by the time it reaches the opponent.

3) d=Vit+1/2gt*2

For this one, let’s give each variable a value. If Viy=5 m/s and t=2 s, dy=5x2-4.9x4=-9.6 m. If Viy is increased to 6 m/s, which means t is reduced let’s say to 1.5 s, then dy=6x1.5-4.9x2.25=-2.025 m. Height is decreased when there is a greater initial velocity and when the projectile happens over a shorter time.

The higher initial velocity you hit the ball with, the lower a maximum height is yielded, leading to a greater velocity the opponent receives.

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