Question

A particle has such a curvilinear motion that its x coordinate is defined by 𝑥 = 5𝑡 3 −105𝑡 where 𝑥 is in inches and 𝑡 is in seconds. When 𝑡 = 2 𝑠, the total acceleration is 75 𝑖𝑛/𝑠 2 . If the y component of acceleration is constant and the particle starts from rest at the origin when 𝑡 = 0, determine its total velocity when 𝑡 = 4 𝑠.

Answer: 𝒗 = 𝟏𝟖. 𝟕𝟓 𝒇𝒕/s

Answer 1

Total acceleration , $\vec{a}=a_x{i}+a_y{j}=30t{i}+a_y{j}$

$\vec{a}= \sqrt{900t^2+a_y^2}; t=2 \implies75=\sqrt{900*2^2+a_y^2} \implies a_y=45 in/s^2$

$\vec{a}=[30t{i}+45{j}] in/s^2$

Velocity at t=4 sec $\implies \vec{v}=\vec{u}+\vec{a}t=-105i+ [(30*4){i}+45{j}]*4$

$\vec{v}=[375{i}+180{j}]in/s$

$\vec{|v|}=\sqrt{375^2+180^2}=415.96 in/s=415.96in/s^2=34.66 ft/s^2$