Question

A particle, originally at rest and located at point (3 ft, 2 ft, 5 ft), is subjected to an acceleration a = {6ti + 12t^2k} ft/s^2. Determine the particle’s position (x, y, z) when t = 2 s.

Answer 1

Answer:

The particle's position at t = 2 is

r = (11ft, 2ft, 21ft) = (11, 2, 21) ft

Explanation:

r₀ = (3, 2, 5) ft

a = (6t, 0, 12t²) ft/s²

a = dv/dt

dv/dt = (6tî + 0j + 12t²ķ)

dv = (6tî + 0j + 12t²ķ) st

Integrating the left hand side from 0 to v (the particle was originally at rest) and the right hand side from 0 to t,

We obtain,

v = (3t²î + 0j + 4t³ķ) ft/s

v = dr/dt

dr/dt = (3t²î + 0j + 4t³ķ)

dr = (3t²î + 0j + 4t³ķ) st

Integrating the left hand side from r₀ (the original position) to r and the right hand side from 0 to t,

r - r₀ = (t³î + 0j + t⁴ķ) ft

r = (t³î + 0j + t⁴ķ) + r₀

r = (t³î + 0j + t⁴ķ) + (3î + 2j + 5ķ)

At t=2s, t³ = 8 and t⁴ = 16

r = (8î + 0j + 16ķ) + (3î + 2j + 5ķ)

r = (11î + 2j + 21ķ) ft

r = (11ft, 2ft, 21ft)

Hope this helps!