Refer to Figure 1. Indicate whether the magnitude of the cart's average velocity measured by the runner in Scenario 2, $\left|\overline{v}_{c,r}\right|$, is greater than, less than, or equal to the magnitude of the cart's average velocity measured by the student on the ground in Scenario 1, $\left|\overline{v}_{c,g}\right|$, by writing one of the following in your answer booklet.

Starting with a fundamental definition of average velocity and the definition of relative position, derive an expression for the cart's average velocity relative to the runner, $\overline{\vec{v}}_{c,r}$, in terms of $\vec{v}_{c,g}$ and $\vec{v}_{r,g}$. Express your final answer as a one-dimensional vector equation that includes appropriate direction (sign). Begin your derivation by writing a fundamental physics principle or an equation from the reference information. Consider the general case in which an object (the cart) moves with constant velocity $\vec{v}_{c,g}$ relative to the ground, and an observer (the runner) moves with constant velocity $\vec{v}_{r,g}$ relative to the ground. Both velocities are along the same straight line (east-west), and east is the positive direction.

Indicate whether the expression you derived in part B is or is not consistent with the claim made in part A. Briefly justify your answer by referencing how the magnitudes compare when $\vec{v}_{c,g} = 2.0\ \text{m/s east}$ and $\vec{v}_{r,g} = 1.5\ \text{m/s east}$, and by stating the direction of $\overline{\vec{v}}_{c,r}$.