Question

A 0.30 kg football is thrown from a height of 2.5 m with an initial velocity of 25.0 m/s (34° above the horizontal axis). Assuming that there is no air resistance and that the ball lands on the ground, how far will the ball travel horizontally?

Answer 1

Given:

The mass of the football is,

$m=0.30\text{ kg}$

The initial height of the football is,

$h=2.5\text{ m}$

The initial velocity of the football is,

$u=25.0\text{ m/s}$

The angle with the horizontal is,

$\theta=34\degree$

To find:

how far will the ball travel horizontally

Explanation:

The horizontal component of the initial velocity is,

$\begin{gathered} u_x=ucos\theta \\ =25.0\times cos34\degree \\ =20.7\text{ m/s} \end{gathered}$

The vertical component of the initial velocity is,

$\begin{gathered} u_y=usin\theta \\ =25.0\times sin34\degree \\ =13.9\text{ m/s} \end{gathered}$

Applying the equation of motion along the vertical direction we get,

$\begin{gathered} y=u_yt-\frac{1}{2}gt^2 \\ 1.8=13.9t-\frac{1}{2}\times9.8\times t^2 \\ 4.9t^2-13.9t+1.8=0 \\ t=\frac{13.9\pm\sqrt{(-13.9)^2-4\times4.9\times1.8}}{2\times4.9} \\ t=\frac{13.9\pm12.6}{9.8} \\ t=2.70\text{ s, 0.13 s} \end{gathered}$

The horizontal distance covered by the football in this time is,

$\begin{gathered} d=u_xt \\ =20.7\times2.70 \\ =55.9\text{ m} \\ or \\ d=20.7\times0.13 \\ =2.69\text{ m} \end{gathered}$

Hence, the ball travels horizontally 55.9 m or 2.69 m.