Question

Now Abel and Kato use what they learned to answer the following problem. The initial speed of a tennis ball is 54 m/s and the launch angle is θi = 21°. Neglect air resistance. What is the maximum height, h, of the tennis ball? m What is the range, R, of the tennis ball?

Answer 1

Answer:

h=19.4m

R=199.07 m

Explanation:

To solve this problem we use the parabolic motion equations:

We define:

$v_{i}$total initial speed  =54$\frac{m}{s}$

$\alpha$=angle that forms the total initial speed with the horizontal line= 21°

$v_{ix}$:initial speed component in horizontal direction

=$v_{i} cos\alpha$ =54*cos 21= 50.41 m/s

$v_{iy}$:initial speed component in vertical direction

=$v_{i} sin\alpha$=54*sin21=19.35 m/s

$v_{x}$ :horizontal speed at any point on the parabolic path

$v_{y}$: vertical speed at any point on the parabolic path

g= acceleration of gravity= 9,8$\frac{m}{s^{2} }$  

Equation of the speed of the football in the vertical direction :

$(v_{y} )^{2} =(v_{yi} )^{2} -2*g*y$  Equation (1)

Calculation of the maximum height(h)

The speed of the ball (vy) in the vertical direction gradually decreases until its value is zero when it reaches the maximum height.

We replace y=h, $v_{y} =0$,  $v_{iy} = 19.39\frac{m}{s}$, $g=9.8\frac{m}{s^{2} }$ in the equation(1)

$0=19.35^{2} -2*9.8*h$

$h=\frac{19.35^{2} }{2*9.8}$

h=19.1 m

Calculating of the range (R)

Fórmula: $R=v_{ix} *t (m)$   Equation (2)

R is the maximum horizontal distance the ball reaches.

The time (t) for the ball to reach R is twice the time the ball spends to reach the maximum height. Then, we calculate the time$(t_{h} )$) when the ball reaches the maximum height

We apply the following equation to calculate $t_{h}$:

$v_{y} =v_{iy} -g*t_{h}$

$0=19.35-9.8*t_{h}$

$9.8*t_{h} =19.35$

$t_{h} =\frac{19.35}{9.8}$

$t_{h} =1.97 s$

$t=2*t_{h}$

$t=3.95 s$  

We replace $v_{ix} =50.41 \frac{m}{s}$ and $t=3.95\frac{m}{s}$ in the equation (2)

R=50.41*3.95

R=199.07 m