Question

part (a) what is the velocity of the ball, in meters per second, when it reaches the top, or highest point, of its trajectory? (v_{top}=0.000 m/s) correct! part (c) enter an expression for the height of the ball as a function of time in terms of the initial velocity (v_i) and the acceleration (a) and the elapsed time (t). (h(t)=v_it + 1/2at^{2}) correct! part (d) what is the maximum height the ball reaches in meters? (h_{max}=3.265 m) correct! part (e) enter an expression for the elapsed time it takes for the ball to travel from the ground to a given height (h), and the acceleration (a) that height (v_h), and the initial velocity (v_i), the velocity the ball has at (delta t = v_i - v_h/a) incorrect! part (f) how long does it take the ball to reach its maximum point (t_{top}) in seconds? (t_{top}=0.8160 s) correct! part (g) what is the total displacement, in meters, of the ball for its entire motion: traveling from the ground to the top and then returning to the ground? (delta y_{total}=0.000 m) correct! part (h) enter an expression for the total time of flight of the ball: the time from when it is launched to when it lands back on the ground. express this time in terms of (v_i) and

Explanation:

Step1: Analyze motion - equation for height

The height of an object in vertical - motion under constant acceleration is given by the kinematic equation $h(t)=v_it+\frac{1}{2}at^{2}$, where $v_i$ is the initial velocity, $a$ is the acceleration, and $t$ is the time.

Step2: Find time to reach maximum height

At the maximum - height, the final velocity $v_f = 0$. Using the kinematic equation $v_f=v_i + at$, we can solve for the time $t$ to reach the maximum height. Rearranging gives $t=\frac{v_f - v_i}{a}=\frac{0 - v_i}{a}=-\frac{v_i}{a}$. The time to reach the maximum height from the ground is $t_{up}=-\frac{v_i}{a}$.

Step3: Find total time of flight

The motion is symmetric. The total time of flight $t_{total}$ from launch to landing is twice the time it takes to reach the maximum height. So $t_{total}=-\frac{2v_i}{a}$.

Answer:

$t_{total}=-\frac{2v_i}{a}$