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Feature: Physics of throwing
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A new angle on throwing
A long throw-in can be a powerful weapon in football. As the World Cup gets under way, Nick Linthorne explains the physics behind the perfect throw-in
It’s Sunday 25 June 2006 and we are five minutes into the first half of the England versus Germany match. The ball goes out of play not far from the Germangoalline. Gary Neville sends a long throw-in over to the far post, Michael Owen gets a touch and it’s a goal. England go 1–0 up! As this hypothetical situation illustrates, a long throw-in – which involves launching the ball with both hands from behind the head – can make a big difference to a game of football. Neville is the long-throw specialist for the England team: he can dispatch the ball some30–40 m from the sideline and provide his team mates with a great opportunity to score. But how can a player maximize the range of his or her throw? In other words, at what angle should the player throw the ball to make it travel as far as possible? Solving the equation of motion for a projectile yields an answer of 45°. But watch a footballer take a long throw-in and you will notice that the ballis often launched at nearer to 30°. The reason for the disparity lies in the biomechanical structure of the human body.

considerably faster at lower angles. This is because the arrangement of muscle levers in the arms and back allows you to exert more throwing force in the horizontal direction than in the vertical direction. To convince yourself of this, try doing a throw-in at a range ofangles. You will find that it is much more difficult to throw a ball straight up than it is to throw a ball straight ahead. The first step in calculating the optimum launch angle was to obtain a mathematical expression relating release velocity to release angle: 2(Fo – aθ)l m where v is the release velocity, Fo the average force exerted by the player for a horizontal projection angle, a is a constant,θ is the release angle, l is the length over which the ball is accelerated, and m is the mass of the ball. The equation was derived using a simple physical model and the constants obtained by fitting the equation to the data. The next step was to insert this expression into the equation for the range of a projectile: v= R= v 2sin2θ 2g

Long-throw specialist England’s Gary Neville throws theball using a relatively flat trajectory to achieve the greatest distance.

The physics of throw-ins The range of a projectile is strongly dependent on the velocity with which it is released, but it also depends on the angle of release (defined with respect to the horizontal). In terms of basic physics, the optimum launch angle is a simple trade-off between the projectile’s vertical velocity, whichincreases the flight time, and horizontal velocity, which increases the distance the projectile travels while airborne. According to most physics textbooks, the optimum angle is 45°. This result is obtained by taking the equation for the range of a projectile in free flight and then either differentiating the equation with respect to launch angle or plotting the range as a function of launch angleand then looking where the peak lies. However, this approach assumes that the launch velocity is independent of the launch angle, something that is often not true for sportspeople. To overcome this limitation, I and an undergraduate student of mine, David Everett, used a video camera and biomechanical-analysis software to measure how a footballer’s throwing speed varies with throwing angle. Wefound that the footballer was able to launch the ball
Physics World June 2006

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 2 2  v sin θ 
2gh

1 2

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Nick Linthorne researches and teaches the physics of sports at Brunel University, Uxbridge, UK, e-mail nick. linthorne@brunel. ac.uk

where R is the range of the projectile, g is the acceleration due to gravity, and h is the height difference between...
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