A quarterback claims that he can throw the football a horizontal distance of 161 m. Furthermore, he claims that he can do this by launching the ball at the relatively low angle of 34.2 ° above the horizontal. To evaluate this claim, determine the speed with which this quarterback must throw the ball. Assume that the ball is launched and caught at the same vertical level and that air resistance can be ignored. For comparison a baseball pitcher who can accurately throw a fastball at 45 m/s (100 mph) would be considered exceptional.

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agaue agaue · Answer 1 · 2019-10-11T06:58:40+02:00

Answer:

The speed with which this quarterback must throw the ball = 41.14 m/s

Explanation:

Let the velocity of throwing be y,

Consider the vertical motion of football,

We have equation of motion, v = u + at

At maximum height final velocity, v = 0

Initial velocity, u = y sin 34.2 = 0.562 y

Acceleration, a = -9.81 m/s²

Substituting,

[tex]0=0.562y-9.81t\\\\t=\frac{0.562y}{9.81}[/tex]

Total time of flight = 2 x Time to reach maximum height

[tex]\texttt{Total time of flight = }\frac{2\times 0.562y}{9.81}=0.115y[/tex]

Consider the horizontal motion of football,

We have equation of motion, s = ut + 0.5at²

Initial velocity, u = y cos 34.2 = 0.827 y

Acceleration, a = 0 m/s²

[tex]\texttt{Total time of flight, t = }0.115y[/tex]

Substituting

[tex]s=0.827y\times 0.115y+0.5\times 0\times (0.115y)^2=0.095y^2[/tex]

This is the maximum horizontal distance he can throw,

Given that

0.095y² = 161

y = 41.14 m/s

The speed with which this quarterback must throw the ball = 41.14 m/s