Refer to Figure 1. Indicate whether the magnitude of the average velocity measured over the first 0.50 s after release is greater in Scenario 1 than in Scenario 2 by writing one of the following in your answer booklet.

Starting with the definition of average velocity, derive an expression for the magnitude of the ground-frame average velocity $|\vec{v}_{\text{avg},1}|$ of the ball over the interval $\Delta t$. Express your answer in terms of $u$, $v_0$, $g$, and $\Delta t$. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. Consider the general case in which, in the cart reference frame, the cart moves at a constant speed $u$ to the right relative to the ground and the ball is thrown straight upward relative to the cart with initial speed $v_0$. Over a time interval $\Delta t$ after release, the acceleration of the ball is $\langle 0, -g \rangle$ in both inertial frames.

Indicate whether the expression you derived in part B is or is not consistent with the claim you made in part A. Briefly justify your answer by referencing how $u$ affects $|\vec{v}_{\text{avg},1}|$ in your expression from part B and comparing it to the cart-frame average velocity over the same interval.