Physics, 03.07.2019 15:30, candycrush24

Two basketball players are essentially equal in all respects. (they are the same height, they jump with the same initial velocity, etc.) in particular, by jumping they can raise their centers of mass the same vertical distance, h (called their "vertical leap"). the first player, arabella, wishes to shoot over the second player, boris, and for this she needs to be as high above boris as possible. arabella jumps at time t=0, and boris jumps later, at time tr (his reaction time). assume that arabella has not yet reached her maximum height when boris jumps. find the vertical displacement d(t)=ha(t)−hb(t), as a function of time for the interval 0

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answered: SsjDavid

The difference in height between both players, Arabella and Boris, is given by D(t) = h_A(t) - h_B (t)

Further explanation

This problem is a 1-dimensional free fall type of problem, it's one dimensional since both players only move in 1 dimension (meaning vertically), and it is a free fall problem since, when both players leave the ground, the only force which is apllied over them is the force excerted by gravity.

To solve our problem, let's suppose that both players jump from a point called the origin, meaning a point where we consider that their vertical displacement is 0 (in this case, the floor would be considered as the origin). Besides that, let's suppose that we start counting the time at the moment when Arabella makes her jump, and let's say that both players jump with velocity V. Therefor from the equations from kinematics about free falling objects we can write an expression for the height of Arabella over time as:

$h_A (t) = V \cdot t - \frac{g \cdot t^2}{2}$

Which is an equation valid since the time Arabella jumps until she gets back on the ground, in math terms this would be written as that the equation is valid over the interval $0\leq t\leq \frac{2 \cdot V}{g}$ . At all other times the height of Arabella would be zero (since she is on the ground).

We can write the same for Boris, though his equation is shifted in time by his reaction time t_r :

$h_B (t) = V \cdot (t-t_r) - \frac{g \cdot (t-t_r)^2}{2}$

Which is valid over the interval $t_r \leq t\leq t_r + \frac{2 \cdot V}{g}$ , and zero everywhere else (since Boris is touching the ground at all other moments). Putting all the pieces together, we can get a piecewise function that expresses the difference in height between both players:

$D(t) = h_A(t) - h_B (t) = V \cdot t - \frac{g \cdot t^2}{2}$ for all $0 \leq t \leq t_r$

$D(t) = h_A(t) - h_B (t) = V \cdot t - \frac{g \cdot t^2}{2} - (V \cdot (t-t_r) - \frac{g \cdot (t-t_r)^2}{2}) = V \cdot t_r - g \cdot t_r \cdot t + \frac{g \cdot t_r^2}{2}$ for all $t_r \leq t \leq \frac{2 \cdot V}{g}$

$D(t) = h_A(t) - h_B (t) = - V \cdot (t-t_r) + \frac{g \cdot (t-t_r)^2}{2}$ for all $\frac{2 \cdot V}{g} \leq t \leq t_r + \frac{2 \cdot V}{g}$

Learn more

Here is a list of well explained free falling problems, similar to yours:

Keywords

Free fall, parabolic motion, kinematics

answered: astultz309459

Vertical displacement when t=t_r is $\b{\boxed{\sqrt{2gH}\left(t\right)-\dfrac{1}{2}g{t^2}}}$ and vertical displacement when ${t_r}$ is $\b{\fbox{\begin\{t({\sqrt{2gH}+\dfrac{g(t_r-2t)}{2})}\end{minispace}}}$ .

Further Explanation:

Arabella and Boris are of same height and jumps with the same initial velocity. Therefore, both can jump to height .

The energy of both the players Arabella and Boris at ground level and height is constant.

Concept:

Applying conservation of energy at ground and height :

$mgH=\dfrac{1}{2}m{v_0}^2$

Rearrange the above equation for initial velocity v_0 :

$v_0=\sqrt{2gH}$

To calculate the height of both players, we can apply the equation of law of motion.

For a given initial velocity v_0 , acceleration , the height of the Arabella in time :

$\boxed{{h_a}\left(t\right) ={h_0}+{v_0}t-\dfrac{1}{2}g{t^2}}$

The reaction time for Boris is t_r . Therefore, to reach the height time taken by her is (t-t_r) .

For a given initial velocity v_0 , acceleration , the height of the Boris in time (t-t_r) :

$\boxed{{h_b}(t)={h_0}+{v_0}(t-t_r)+\dfrac{1}{2}g(t-t_r)^2}$

It is given that Boris jumps before Arabella reaches the maximum height. Therefore, reaction time for Boris cannot be greater than total time.

Case I:

For t=t_r , we get ${h_b}(t)={h_0}$ .

Therefore, the vertical displacement is:

$\begin{aligned}D(t)&={h_a}(t)-{h_b}(t)\\&={h_0}+{v_0}t-\dfrac{1}{2}g{t^2}-{h_0}\\&={v_o}t-\dfrac{1}{2}g{t^2}\\&=\sqrt{2gH}(t)-\dfrac{1}{2}g{t^2}\end{gathered}$

Case II:

For ${t_r}$ , the vertical displacement is:

$\begin{aligned}D(t)&={h_a}(t)-{h_b}(t)\\&={t_r}({\sqrt{2gH}+\dfrac{g(t_r-2t}{2})\end{aligned}$

Thus, vertical displacement when t=t_r is $\b{\boxed{\sqrt{2gH}\left(t\right)-\dfrac{1}{2}g{t^2}}}$ and vertical displacement when ${t_r}$ is $\b{\fbox{\begin\{t({\sqrt{2gH}+\dfrac{g(t_r-2t)}{2})}\end{minispace}}}$ .

Learn more:

1. Average translational kinetic energy:

2. Stress on a body under tension:

3. Water is a compound:

Answer Details:

Grade: High School

Subject: Physics

Chapter: Kinematics

Keywords:

Two, basketball, player, same, height, initial, velocity, vertical, distance, Arabella, Boris, high, above, D(t)=Ha(t) – Hb(t), function, reaction, ground, reaches and time.

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