FalseThe term "arbitrary" refers to an unpredictable direction that cannot be measured or predicted precisely. Rising or decaying oscillations with a certain frequency are referred to as arbitrary motion. (Uniform and non uniform motion)
a) False
Average velocity v_{\text{average}} = \frac{\text{ Total displacement }}{\text{ time }}
The following relation cannot hold all the arbitrary relations since it is arbitrary motion.
v_{\text{average}} = \frac{1}{2} (v(t_1) + v(t_2))
b) True
v_{\text{average}} = \frac{(r(t_2) - r(t_1))}{t_2 - t_1}
This equation can reflect the arbitrary movement of the particle. Since it can be applicable in uniform and non uniform motion.
c) False
v(t) = v(0) + at
This motion equation is only valid for uniform movement. Therefore, this equation cannot represent the movement of the space particle.
d) False
r(t) = r(0) + v(0)t + \frac{1}{2} at^2
The provided equation is only valid for uniform acceleration motion. Since the particle's motion is arbitrary, and its acceleration can also be non-uniform. Therefore, this equation cannot describe particle motion in space.
e) True
a_{\text{average}} = \frac{(v(t_2) - v(t_1))}{t_2 - t_1}
This equation can describe the particle's arbitrary motion. This is the universal relationship between acceleration and velocity-time.
Acceleration: change in velocity with respect to time
a = \frac{\Delta v}{\Delta t}