A particle started from the origin and the components at time t are 2t+3 and 4t.find the path
To find the path of the particle, we need to integrate the components of the particle's position with respect to time.
Given that the particle's components at time t are (2t + 3) and (4t) respectively, we can represent its position as a vector function r(t) = (2t + 3)i + (4t)j.
To find the path, we need to integrate the components of the position vector with respect to time.
Integrating the x-component, we have:
∫(2t + 3) dt = t^2 + 3t + C₁
Integrating the y-component, we have:
∫(4t) dt = 2t^2 + C₂
Therefore, the path of the particle is given by the vector function:
r(t) = (t^2 + 3t + C₁)i + (2t^2 + C₂)j
Where C₁ and C₂ are constants of integration. These constants depend on any initial conditions or constraints that may be given in the problem.