A particle moves in a straight line such that its position in metres from a fixed origin is given by s ( t ) = t 3 - 3 t + 1 , where t is the time in seconds, t ≥ 0 . Find the initial position, velocity and acceleration. Hence describe the particle's motion at this instant.

Given that, a particle moves in a straight line such that its position in metres from a fixed origin is given by s ( t ) = t 3 - 3 t + 1 , where t ≥ 0 [Seconds] We know that, Velocity v t is the derivative of the displacement function s t and Acceleration a t , is the derivative of the velocity v t . i.e., v t = d s d t or s ' t and a t = d v d t = d 2 s d t 2 or s ' ' t . Now, differentiate s t with respect to t , then we get: d s d t = v t = 3 t 2 - 3 Again differentiate v t with respect to t , then we get: d 2 s d t 2 = a t = 6 t Now, the initial position of displacement is s 0 = 1 m And also, the initial position of velocity is v 0 = - 3 m / s Again, the initial position of Acceleration is a 0 = 0 At this instant, the particle is 1 metre from the origin in the positive direction, travelling towards the origin at - 3 m / s , and is not accelerating.

A particle moves in a straight line such that its position in metres from a fixed origin is given by s ( t ) = t 3 - 3 t + 1 , where t is the time in seconds, t ≥ 0 . Find the initial position, velocity and acceleration. Hence describe the particle's motion at this instant.

Given that, a particle moves in a straight line such that its position in metres from a fixed origin is given by s ( t ) = t 3 - 3 t + 1 , where t ≥ 0 [Seconds] We know that, Velocity v t is the derivative of the displacement function s t and Acceleration a t , is the derivative of the velocity v t . i.e., v t = d s d t or s ' t and a t = d v d t = d 2 s d t 2 or s ' ' t . Now, differentiate s t with respect to t , then we get: d s d t = v t = 3 t 2 - 3 Again differentiate v t with respect to t , then we get: d 2 s d t 2 = a t = 6 t Now, the initial position of displacement is s 0 = 1 m And also, the initial position of velocity is v 0 = - 3 m / s Again, the initial position of Acceleration is a 0 = 0 At this instant, the particle is 1 metre from the origin in the positive direction, travelling towards the origin at - 3 m / s , and is not accelerating.