Question

Suppose a soccer player kicks the ball from a distance 29 m toward the goal. find the initial speed of the ball if it just passes over the goal, 2.4 m above the ground, given the initial direction to be 39° above the horizontal.

Answer 1

The ball's vertical velocity at the time it just passes over the goal is 0 m/s. Its initial vertical velocity is unknown and we denote it by $v\sin39^\circ$, where $v$ here is the ball's initial speed. Vertically, the only force acting on the ball is gravity, which attributes a downward acceleration of 9.8 m/s^2. We expect the maximum height achieved by the ball to be 2.4 m, so we can find the initial speed by solving

$\left(0\,\dfrac{\mathrm m}{\mathrm s}\right)^2-\left(v\sin39^\circ\right)^2=2\left(-9.8\,\dfrac{\mathrm m}{\mathrm s^2}\right)(2.4\,\mathrm m)$

$\implies v=11\,\dfrac{\mathrm m}{\mathrm s}$

Answer 2

Initial speed of the ball if it just passes over the goal, 2.4 m above the ground will be 18.16 m/s

What is equation of motion

Equations that describe the behavior of a physical system in terms of its motion as a function of time are  called equation of motion

The branch of physics that defines motion with respect to space and time , ignoring the cause of that motion is known as kinematics. Kinematics equation are a set of equations that can derive an unknown aspect of a body's motion if the other aspects are provided .

let initial speed be = u m/s

Since , it is a projectile motion

speed will have two components = u (x)  horizontal component

                                                           u (y) vertical component

let angle made from ground be = x degree

cos x = u (x) / u

u(x) = u cos 39

sin x = u(y) / u

u(y) = u sin 39

u(x) = distance / time (t)

time (t)= 29 / u(x)

t = 29 / u sin 39

Since , the ball is 24 m above the ground , using equation of motion

s = u t + 1/2 a(t^2)

2.4 = u(y)t + 1/2 (-g) (t^2)                  a (downward ) = -g

substituting the value of u(y)

2.4 = u sin 39 *  t + 1/2 (-9.8) (t^2)

2.4 = u sin 39 * t   -  4.90 (t^2)

substituting the value of t

2.4 = u sin 39 * 29 / u sin 39     -     4.90 ((29 / u sin 39)^2)

2.4 = 29 tan 39  - (4120 / (u^2) * 0.593)

2.4  = 23.48 -  (4120 / (u^2) * 0.593)

-21.08 = - 6949.24/(u^2)

u^2 = 329.66

u= 18.16 m/s

initial speed of the ball will be 18.16 m/s

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