Question

problem 3
a rocket fires two engines simultaneously. one produces a thrust of 725 n directly forward while the other gives a 513-n thrust at 32.4° above the forward direction. find the magnitude and direction (relative to the forward direction) of the resultant force that these engines exert on the rocket.

Explanation:

Step1: Define coordinate system

Forward as x-axis, upward as y-axis.

Step2: Calculate F1 components

F₁=725N (forward), so F₁ₓ=725N, F₁ᵧ=0N.

Step3: Calculate F2 components

F₂=513N, θ=32.4°. F₂ₓ=513cos(32.4°)≈513×0.844≈433N; F₂ᵧ=513sin(32.4°)≈513×0.536≈275N.

Step4: Sum components

Rₓ=F₁ₓ+F₂ₓ=725+433=1158N; Rᵧ=F₁ᵧ+F₂ᵧ=0+275=275N.

Step5: Resultant magnitude

|R|=√(Rₓ²+Rᵧ²)=√(1158²+275²)≈√(1341164+75625)=√1416789≈1190N.

Step6: Resultant direction

θ=arctan(Rᵧ/Rₓ)=arctan(275/1158)≈13.4°.

Answer:

Magnitude: 1190 N, Direction: 13.4° above forward direction.