# How the Physics of Football Works

Punting: Hang-Time, Peak Height and Range

The parabolic path of a football can be described by these two equations:

y = Vyt - 0.5gt2

### x =Vxt

x =Vxt

• y is the height at any time (t)
• Vy is the vertical component of the football's initial velocity
• g is acceleration due to Earth's gravity, 9.8 m/s2
• x is the horizontal distance of the ball at any time (t)
• Vx is the horizontal component of the football's initial velocity

To calculate the hang-time (ttotal), peak height (ymax), and maximum range (xmax) of a punt, you must know the initial velocity (V) of the ball off the kicker's foot, and the angle (theta) of the kick.

The velocity must be broken into horizontal (Vx) and vertical (Vy) components according to the following formulas:

• Vx = V cos(theta)
• Vy = V sin(theta)

The hang-time (ttotal) must be determined by one of these two formulas:

• ttotal = (2Vy/g)
• ttotal = (0.204Vy)

Once you know the hang-time, you can calculate maximum range (xmax):

• xmax = Vx ttotal

You can calculate the time (t1/2) at which the ball is at its peak height:

• t1/2 = 0.5 ttotal

And you can calculate the peak height (ymax), using one of these two formulas:

• ymax= vy(t1/2) - 1/2g(t1/2)2
• ymax = vy(t1/2) - 0.49(t1/2)2

For example, a kick with a velocity of 90 ft/s (27.4 m/s) at an angle of 30 degrees will have the following values:

Vertical and horizontal components of velocity:

• Vx = V cos(theta) = (27.4 m/s) cos (30 degrees) = (27.4 m/s) (0.0.87) = 23.7 m/s
• Vy = V sin(theta) = (27.4 m/s) sin (30 degrees) = (27.4 m/s) (0.5) = 13.7 m/s

Hang-time:

• ttotal = (0.204Vy) = {0.204 (13.7m/s)} = 2.80 s.

Maximum range:

• xmax = Vx ttotal = (23.7 m/s)(2.80 s) = 66.4 m
• 1 m = 1.09 yd
• xmax = 72 yd

Time at peak height:

• t1/2 = 0.5 ttotal = (0.5)(2.80 s) = 1.40 s

Peak height:

• ymax = Vy(t1/2) - 0.49(t1/2)= [{(13.7 m/s)(1.40 s)} - {0.49(1.40 s)2}] = 18.2 m
• 1 m = 3.28 ft
• ymax = 59.7 ft

If we do the calculations for a punt with the same velocity, but an angle of 45 degrees, then we get a hang-time of 3.96 s, a maximum range of 76.8 m (84 yd), and a peak height of 36.5 m (120 ft). If we change the angle of the kick to 60 degrees, we get a hang-time of 4.84 s, a maximum range of 66.3 m (72 yd), and a peak height of 54.5 m (179 ft). Notice that as the angle of the kick gets steeper, the ball hangs longer in the air and goes higher. Also, as the angle of the kick is increased, the distance traveled by the ball increases to a maximum (achieved at 45 degrees) and then decreases.