Final answer:
The question involves Physics concepts related to the motion of a particle on the x-axis, including finding its velocity, acceleration, and position at various times using kinematic equations and integration or differentiation of the velocity function when necessary.
Step-by-step explanation:
The student's question is related to the motion of a particle along the x-axis and involves finding various components of the particle's motion such as its position, velocity, and acceleration at different points in time. When a particle moves along the x-axis, and its velocity is given as a function of time, you may need to use kinematic equations to analyze the motion.
For a particle with a velocity functions v(t) = A + Bt¯¹ and the given initial conditions, to find the acceleration at t = 2.0 s and t = 5.0 s, you would differentiate the velocity function with respect to time. To find the position at those times, you would integrate the velocity function from the initial time to the given time.
Moreover, when a particle's position is given as x(t) = 4.0 - 2.0t m, this linear equation suggests the particle is moving along the x-axis with a constant velocity. The time at which the particle crosses the origin can be found by setting x(t) to zero and solving the equation for t. The displacement between two points in time is found by subtracting the initial position from the final position.
Lastly, for a particle moving with constant acceleration, you can use the kinematic equation x(t) = x_0 + v_0t + ½at² to determine its position at any given time t, given the initial position x_0, initial velocity v_0, and constant acceleration a.