using vectors to solve speed and time and height

Question

If Orion wants to jump a motorcycle a distance of 400 feet, what is the minimum speed he will have to go to be able to make the jump? How long will he be in the air? What will the highest point be that he gets to on his motorcycle?

 

Answer 1

v=v₀cosθ*i + v₀sinθ*j

X=v₀cosθ*t = 400 ft and Y=v₀sinθ*t - gt²/2;

when Y=0; so v₀sinθ*t = gt²/2; then X=400=v₀cosθ*t, so t=X/v₀cosθ,

from v₀sinθ = gt/2; v₀sinθ = g/2 *X/v₀cosθ; v²₀ 2sinθ cosθ = X/g; v²₀ = Xg sin2θ;

v₀ min is for sin2θ = max = 1; so v₀ min = [Xg]^1/2 V_{0 min} = [400*32.17]^1/2 = 113.44 ft/sec

t=X/v₀cosθ = 400/113.44*cos(45) = 4.99 sec.

Ymax is when V_y=V₀sinθ - gt =0; so t=V₀sinθ/g;

Ymax = V₀sinθ*t - gt²/2 = V₀sinθ [V₀sinθ/g] - g/2 [V₀sinθ/g]^{2 =} [V₀sinθ]²/2g

Ymax= [113.44 *sin(45)]²/2*32.17 = 100 ft.