The oblique throw and the laboratory work

We split the initial velocity of a body thrown at angle α to the horizontal with speed υ₀ into two components: υ₀ₓ = υ₀cosα and υ₀ᵧ = υ₀sinα. Horizontally the body moves uniformly (υₓ = υ₀cosα, x = υ₀t·cosα); vertically it moves with uniform deceleration from the initial speed υ₀sinα (υᵧ = υ₀sinα – gt). At the highest point υᵧ = 0, so the speed there is the smallest, υ = υ₀cosα. The rise time is tₖ = υ₀sinα/g, the total flight time is t = 2υ₀sinα/g, the maximum height is h = υ₀²sin²α/(2g), and the range is s = υ₀²sin2α/g. Without air resistance the range is greatest at 45° and the same for angles α and 90° – α. The trajectory equation is y = x·tanα – gx²/(2υ₀²cos²α), a parabola through the origin opening downward. In real conditions air resistance shortens the range and the best angle is below 45° (roughly 30–40°).