Irodov Problem 1.199

May 19, 2026

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Problem Statement

A ball is thrown at speed v₀ at angle θ₀ above the horizontal. Show that the trajectory is a parabola and find the range R, maximum height H, and time of flight T.

Given

v₀, θ₀, g. Projectile motion.

Concepts & Formulas

x = v₀cosθ₀·t, y = v₀sinθ₀·t − ½gt². Eliminate t to get y(x).

Step-by-Step Solution

Step 1: t = x/(v₀cosθ₀).
Step 2: y = x·tanθ₀ − gx²/(2v₀²cos²θ₀) — parabola.
Step 3: H = v₀²sin²θ₀/(2g).
Step 4: T = 2v₀sinθ₀/g.
Step 5: R = v₀²sin2θ₀/g.

Worked Calculation

H = v₀²sin²θ₀/(2g). T = 2v₀sinθ₀/g. R = v₀²sin2θ₀/g.

Boxed Answer

H = v₀²sin²θ₀/(2g); T = 2v₀sinθ₀/g; R = v₀²sin2θ₀/g

Physical Interpretation

Range is maximum at θ₀ = 45°. The parabolic form arises because horizontal motion is uniform while vertical motion is uniformly accelerated.