Question

A faulty model rocket moves in the xy-plane (the positive y-direction is vertically upward). The rocket's acceleration has components ax(t)=αt2and ay(t)=β−γt, where α = 2.50 m/s4, β = 9.00 m/s2, and γ = 1.40 m/s3. At t=0 the rocket is at the origin and has velocity v⃗ 0=v0xi^+v0yj^with v0x = 1.00 m/s and v0y = 7.00 m/s.

A. Calculate the velocity vector as a function of time.

Express your answer in terms of v0x, v0y, β, γ, and α. Write the vector v⃗ (t) in the form v(t)x, v(t)y, where the x and y components are separated by a comma.

B. Calculate the position vector as a function of time.

Express your answer in terms of v0x, v0y, β, γ, and α. Write the vector r(t)→ in the form r(t)x, r(t)y where the x and y components are separated by a comma.

D. Sketch the graph

E. What is the horizontal displacement of the rocket when it returns to

y=0?

Answer 1

a(x,y) = alpha t^2 i + (beta - gamma)t j

we know that

a = vd/dt => dv = a dt

dv = [2.5 t^2 i + (9 - 1.4)t j] dt

dv = (alpha t^2 i + (beta - gamma t) j

integrating this we get

v = (alpha t^3/3 + v0x )i + (beta t - gamma t^2/2 + voy) j

V = 2.5/3 t^3 + v0x + 9 t - 1.4/2 t^2 + v0y

Vx,Vy = (alpha t^3/3 + v0x ),(beta t - gamma t^2/2 + voy)

V = (0.833 t^3 + 1)i + (9 t - 0.7t^2 + 7) j m/s

B)v = dr/dt => dr = v dt

dr = [(alpha t^3/3 + v0x )i + (beta t - gamma t^2/2 + voy) j]

r = (alpha t^4/12 + v0x t + r0x)i + beta t^2/2 - gamma t^3/6 + voy t + r0y) j

rx,ry = (alpha t^4/12 + v0x t + r0x), beta t^2/2 - gamma t^3/6 + voy t + r0y)