Question

Imagine you are a punter on a football team and the coach tells you to give your players a chance to get down the field. With which angle with the horizontal would you punt the ball? Explain. A. 15 degrees B. 45 degrees C. 65 degrees D. It doesn't matter.

Answer 1

Coach is telling punter to give other players a chance to get down the field, which means your range of projectile should be maximum. Now If you remember from projectile motion, Range is given by:

R = (V0^2*sin 2)/g

Now this range will be maximum when sin 2 have it's maximum value, which is 1.

Since sine function values lies between -1 and 1, So

sin 2 = 1

= (1/2)*arcsin (1) = (1/2)*90 deg = 45 deg

So you should punt the ball at 45 deg

Correct option is B.

If you need derivation of above expression:

Range is given by:

Range = V0x*T

V0x = Initial horizontal velocity of projectile = V0*cos

Since there is no acceleration in horizontal direction, So horizontal velocity remains constant

T = time taken by ball to return to ground

Using 2nd kinematic equation in vertical direction

h = V0y*T + (1/2)*ay*T^2

V0y = Initial Vertical velocity = V0*sin

ay = acceleration in vertical direction = -g

h = vertical displacement = 0

So,

0 = V0*T*sin - (1/2)*g*T^2

V0*T*sin = (1/2)*g*T^2

T = (2*V0*sin )/g

So,

R = (V0*cos )*((2*V0*sin )/g)

R = (V0^2*2*sin*cos )/g

From trigonometry

2*sin *cos = sin 2

R = (V0^2*sin 2)/g

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