An archer is shooting at targets. The height of the arrow is 5 feet above the ground. Due to safety rules, the archer much aim the arrow parallel to the ground.

How long does it take for the arrow to hit a target that is 3 feet above the ground? Round your answer to the nearest hundredth.

Answer:

The time it takes to hit a target that is 3 feet above ground is approximately 0.35 seconds

Step-by-step explanation:

The given parameters are;

The height of the arrow above the ground = 5 ft.

The height of the target above the ground = 3 ft.

The time it takes the arrow to fall to 3 ft. from 5 ft. above the ground is given by the free fall formula, since the arrow has no vertical velocity component

The equation for free fall is h = 1/2·g·t²

Where;

h = The height of fall = 5 - 3 = 2 feet

g = The acceleration due to gravity = 32.17405 ft/s²

t = How long it takes to fall (to its target)

∴ 2 = 1/2 × 32.17405 × t²

t² = 2/(1/2 × 32.17405) ≈ 0.1243

t = √0.1243 ≈ 0.353

t = 0.35 seconds, to the nearest hundredth

Therefore, the time it takes to fall 2 ft to hit a target that is 3 feet above ground = 0.35 seconds.

The time to hit the target is simply the period the arrow gets to the target from the arch.

The time to hit the target is 0.35 seconds

The height of a parabola is calculated as:

[tex]\mathbf{h = \frac{1}{2}gt^2}[/tex]

Where:

[tex]\mathbf{h =5 - 3 = 2}[/tex] --- the difference between the height of the arrow and the height of the target.

[tex]\mathbf{g = 32.2 fts^{-2}}[/tex] --- acceleration due to gravity

So, we have:

[tex]\mathbf{2 = \frac{1}{2} \times 32.2 \times t^2}[/tex]

Multiply both sides by 2

[tex]\mathbf{4 = 32.2 \times t^2}[/tex]

Divide both sides by 32.2

[tex]\mathbf{0.124 = t^2}[/tex]

Take square roots

[tex]\mathbf{0.35 = t}[/tex]

Rewrite as:

[tex]\mathbf{t = 0.35s}[/tex]

Hence, the time to hit the target is 0.35 seconds

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