Question

A catapult is tested by Roman legionnaires. They tabulate the results in a papyrus and 2000 years later the archaeological team reads (distances translated into modern units): Range = 0.4 km; angle of launch = π/5 rad; landing height = launch height. What is the initial velocity of launch of the boulders if air resistance is negligible?

Answer 1

64.2 m/s

Step-by-step explanation:

In the x direction:

x = x₀ + v₀ₓ t + ½ at²

400 m = 0 m + v₀ cos (π/5) t + ½ (0 m/s²) t²

t = 400 / (v₀ cos (π/5))

In the y direction:

y = y₀ + v₀ᵧ t + ½ gt²

0 m = 0 m + v₀ sin (π/5) t + ½ (-9.8 m/s²) t²

0 = v₀ sin (π/5) - 4.9 t

t = v₀ sin (π/5) / 4.9

Therefore:

400 / (v₀ cos (π/5)) = v₀ sin (π/5) / 4.9

1960 = v₀² sin (π/5) cos(π/5)

1960 = ½ v₀² sin(2π/5)

3920 / sin(2π/5) = v₀²

v₀ = 64.2 m/s

Answer 2

Initial velocity = 423.08m/s

Step-by-step explanation:

Using formular for Range of projectile

R =V^2 sin2theta/g

Given:

Range=0.4km= 400m

Theta=3.142/5

g= 9.8m/s^2

400= V^2×sin(2×3.142/5)/9.8

400×9.8= V^2Sin 1.26

3920= 0.0219V^2

V^2= 3920/0.0219

V^2= 178995.43

V=sqrt 178995.43

V= 423.08m/s