Motion in one dimension

Question

An object is traveling in a straight line. Its acceleration is given by : a = Ctn where C is a constant, n is a real number, and t is time. Find the general equations for the position and velocity of the object as a function of time?

Answer 1

To find the position and velocity of the object as a function of time, we need to integrate the acceleration with respect to time.

Acceleration, a = Ctn

Integrating with respect to time, we get:

∫a dt = ∫Ctn dt

Integrating both sides, we get:

v = (C/2) t^2 + v₀

where v₀ is the constant of integration, which represents the initial velocity.

Integrating velocity with respect to time, we get:

∫v dt = ∫[(C/2) t^2 + v₀] dt

s = (C/6) t^3 + v₀t + s₀

where s₀ is the constant of integration, which represents the initial position.

Therefore, the general equations for the position and velocity of the object as a function of time are:

s = (C/6) t^3 + v₀t + s₀

v = (C/2) t^2 + v₀

where C, n, v₀, and s₀ are constants determined by the initial conditions of the object's motion.