For any arbitrary motion in space, which of the following relations are true: (The ‘average’ stands for average of the quantity over the time interval t1 to t2)
v_average = 1/2 (v(t1) + v(t2))
v_average = [r(t2) - r(t1)] / (t2 - t1)
v(t) = v(0) + a t
r(t) = r(0) + v(0) t + 1/2 a
a_average = [v(t2) - v(t1)] / (t2 - t1)
Explanation
For arbitrary motion in space, we need to identify which relations are universally true based on definitions.
Option (B) is TRUE: The average velocity is defined as the ratio of displacement to the time interval. From the context: 'The average velocity (v̄) of an object is the ratio of the displacement and the corresponding time interval: v̄ = Δr/Δt.' This definition holds for any arbitrary motion.
Option (E) is TRUE: The average acceleration is defined as the change in velocity divided by the time interval. From the context: 'If the velocity of an object changes from v to v' in time Δt, then its average acceleration is given by: ā = (v - v')/Δt = Δv/Δt.' This definition is universally valid.
Options (A), (C), and (D) are FALSE for arbitrary motion: These relations are only valid for uniformly accelerated motion (constant acceleration). The context mentions 'In terms of components: v_x = v_ox + a_x t' and discusses average velocity as '(v₀ + v)/2' specifically for the case of constant acceleration, not arbitrary motion. For arbitrary motion where acceleration may vary with time, these kinematic equations do not hold.