Question

The velocity of particle at any time t, is given by the vector function; v(t) = (4t + 5)i + (3 + t)j. Find the displacement x(t) and the acceleration a(t) at any time, t, if x(4) = 4i - 6j

Answer 1

$$\vec{x}(t)=\left (4\frac{t^2}{2} + 5t+c_1\right)\vec{i} +\left (3t +\frac{t^2}{2}+c_2\right)\vec{j}\\\vec{x}(4)=\left (4\frac{4^2}{2} + 5(4)+c_1\right)\vec{i} +\left (3(4) +\frac{4^2}{2}+c_2\right)\vec{j}\\=\left (4\right)\vec{i} +\left (-6\right)\vec{j}\\c_1=-48,c_2=-26$$

Thus,

$$\vec{x}(t)=\left (2t^2 + 5t-48\right)\vec{i} +\left (3t +\frac{t^2}{2}-26\right)\vec{j}$$

$$\vec{a}(t)=4\vec{i} +\vec{j}$$