A character in a game has an initial velocity of 20.0 m/s at 15.0°. At this point, the character is 25.0 m (horizontally) from the landing platform and the character is 5.00 m high in the air.

Question

A character in a game has an initial velocity of 20.0 m/s at 15.0°. At this point, the character is 25.0 m

(horizontally) from the landing platform and the character is 5.00 m high in the air. Does the character make the jump (land on the platform)? Assume no air resistance. The platform is at height y = 0 m. Assume that the character will land on the platform if the character lands at a distance ≥ 25 m. (In other words, the platform has some length).

(a) Write what is given and what you are trying to find.

(b) Draw a diagram of the motion showing your coordinate system. Be sure to note any distances or angles.

(c) You may start with the projectile motion formulas. Determine whether the game character lands on the platform. Write out all steps (general equation, substitution, algebra, result).

Answer 1

Projectile motion in this case has the formula (x, y) = (v₀t cosθ, y₀ + v₀t sinθ – gt²/2)

with only time t being unknown.

Eliminate t for x to get y = y₀ + x tanθ – (g/2v₀²cos²θ)x².

Therefore, when the character lands at y → 0, their horizontal position x will be the positive root of

0 = x² – (2v₀²sinθcosθ/g)x – 2v₀²y₀cos²θ/g

= (x – v₀²sinθcosθ/g)² – v₀⁴sin²θcos²θ/g² – 2v₀²y₀cos²θ/g

= (x – v₀²sinθcosθ/g)² – (v₀²sin²θ/g + 2y₀)v₀²cos²θ/g.

It's interesting to note from this formula that if the initial height y₀ < –v₀²sin²θ/2g there could be no solution.