Question

part (k) express the velocity of the ball in the instant of time before it hits the ground, (v_f), in terms of (v_i), (t_{total}), and (a). (v_f=(a t_{total}) - v_i) x incorrect! feedback: is available. part (l) what is the velocity of the ball, in meters per second, in the instant of time when it returns to the ground? (v_f=)

Explanation:

Step1: Use the kinematic - equation

The kinematic equation $v_f=v_i + at$. In the vertical - motion of an object under free - fall (assuming no air - resistance), when the object returns to the same height from which it was thrown, the displacement $y - y_0 = 0$. The kinematic equation $y - y_0=v_it+\frac{1}{2}at^{2}$ can be used, but a more straightforward approach is to use $v_f=v_i+at$. The time of flight $t = t_{total}$, and the acceleration $a=-g$ (taking the upward direction as positive). When the ball returns to the starting height, using $v_f=v_i + at$, we substitute $t = t_{total}$ and $a=-g$.

Step2: Analyze the motion

If the ball is thrown upwards with an initial velocity $v_i$ and the acceleration due to gravity is $a=-g$, and the time of flight is $t_{total}$, then $v_f=v_i - gt_{total}$. In the general case of vertical motion with constant acceleration $a$, the final velocity $v_f$ in terms of initial velocity $v_i$, acceleration $a$, and time $t_{total}$ is given by the kinematic equation $v_f=v_i+at_{total}$.

For the case of the ball returning to the ground (same height as the starting point), if we consider the motion under the influence of gravity $a=-g$ (where $g$ is the acceleration due to gravity), and we know that the time of flight is $t_{total}$.
The kinematic equation for velocity $v = v_0+at$ gives $v_f=v_i - gt_{total}$ (where $a=-g$). But if we just use the general form with $a$ as the acceleration, the answer for part (k) is $v_f=v_i+at_{total}$.
For part (l), when the ball returns to the ground, in the absence of air - resistance, the magnitude of the final velocity is the same as the initial velocity but the direction is opposite. So $v_f=-v_i$ (because the motion is symmetric in the vertical direction under constant acceleration and when the starting and ending heights are the same).

Answer:

Part (k): $v_f=v_i + at_{total}$
Part (l): $v_f=-v_i$