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Jash Jobalia Mathematics has applications in many areas, from calculating revenue to artificial intelligence. Today, this article will talk about how mathematics can be used in the sport of table tennis. I have been playing table tennis for the past nine years, and throughout my journey I struggled with one specific shot- the lob shot (this shot is used to receive the smash shot). Therefore, I decided to use mathematics to help me decide the perfect distance from the table tennis table at which the lob shot should be hit. For these calculations there are three variables: Smash shot speed = x ms^-1 Angle at which the smash shot is hit = θ° Height at which smash shot is hit = H metres As we know the smash shot is hit at velocity x ms^-1 and angle θ°, we can find the vertical and horizontal velocity vectors of the ball by using simple trigonometric ratios, forming a right-angled triangle to do this since the table tennis table is flat. The above diagram shows the vertical and horizontal components of velocity as the smash shot is initially hit: vertical component = xcosθ and horizontal component = xsinθ. The above diagram shows the whole journey of the ball where the yellow circles represent the ball. The ball is smashed by the opponent at the point Y1, from where it travels to point Y2 where it hits the table. The ball bounces of the table and starts moving upwards till it reaches a peak at point Y3, the optimum point from where a plyer should hit the lob shot. The ball then hits the racket and moves in the trajectory shown by the orange line till it reaches point Y4, a point anywhere on the table. Now that we have understood the journey of the ball, we can start making our calculations. We need to first find the time taken by the smash shot to reach the table (from point Y1 to Y2). We will call this value of time t0 and this can be found by the kinematics equation s = ut + ½at^2, where s is vertical displacement, u is our initial vertical velocity xcosθ, and a is acceleration due to gravity 9.81 ms^-2. Since we have defined down as the positive direction (as the initial velocity is downwards), this acceleration must be positive. This gives us the following equation. Rearranging this and applying the quadratic formula gives where we will use the positive value of t0. We can use another kinematics equation v^2 = u^2 + 2as to find the velocity of the ball at Y2, where v is the vertical velocity at Y2, u is the vertical velocity at Y1 when the shot is hit (xcosθ), a is acceleration due to gravity and s is the displacement we found above. So, if we define the new vertical velocity component at Y2 to be V0, the following equation holds: This equation lets us find V0 in terms of our initially defined variables x, θ and H, which will be useful later. At point Y2 the ball hits the table and continues moving with the same velocity after it bounces, since momentum is conserved and the mass of the ball does not change. This means the ball will move upwards after point Y2. We can define the optimum height from which the shot should be hit as S0, which is the point where the ball’s vertical velocity is 0. This makes it easier for the player to judge how to hit the shot. From here we will calculate the time (t1) it will take for the ball to reach a displacement of S0 at point Y3 where the vertical speed of the ball is zero. The vertical speed of the ball becomes zero as the ball is affected by gravity on its way up. However, to find t1 we must first find S0 by the equation v^2 = u^2 + 2as. v is 0 since the vertical velocity is 0 at point Y3; u is -V0 because it has the same magnitude as the velocity of the ball as it hits Y2 but is moving upwards so the sign is flipped; a is 9.81 ms^-2. S0 must also be negative because it is an upwards displacement. Plugging in these values gives the following: t1 can be found using s = ut + ½at^2, where u is -V0; t is t1; a is 9.81 ms^-2. Substituting in our S0 from above and rearranging gives us the following quadratic equation: so t1 = V0 / 9.81. We know from above that (V0)^2 = (xcosθ)^2 + 19.62H, so we can substitute this into the above equation, yielding We have now calculated t1 which we will use along with t0 to find the total horizontal distance the ball has travelled from point Y1 to Y3. Since the horizontal component of speed has remained constant throughout as we assumed there was no air resistance, we can simply use the formula distance = speed * time to calculate the horizontal distance travelled by the ball (D0) from Y1 to Y3. This can be given by the expression: We can substitute in our expressions for t0 and t1 to obtain D0 in terms of our initial variables: We can now subtract the length of the table (2.74m) from D0 to give the distance in metres the player should stand from the table to hit the optimum lob shot. (D1) This shows us some of the ways that mathematics can be used to help sports players. You can find a more detailed analysis of the maths involved in the table tennis lob shot here. For suggestions or feedback, please write to [email protected].
7 Comments
20/12/2022 01:07:58 am
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5/1/2023 03:43:16 am
100 tl deneme bonusu veren siteleri öğrenmek istiyorsan tıkla.
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