A) The balloon hits the ground after t = 2.55 sec
B) Horizontal distance covered: 22.5 m
C) Horizontal distance covered: 15.9 m
Explanation:
A)
The motion of the balloon is a projectile motion, so it consists of a uniformly accelerated motion along the vertical direction and of a uniform motion along the horizontal direction.
The vertical motion can be analyzed using the suvat equation:
[tex]s=u_y t - \frac{1}{2}gt^2[/tex]
where:
s = -9.5 m is the vertical displacement of the balloon when it reaches the ground
[tex]u_y = u sin \theta[/tex] is the initial vertical velocity of the balloon, where
u = 12.5 m/s is the initial speed
[tex]\theta=45^{\circ}[/tex] is the angle of projection
[tex]g=9.8 m/s^2[/tex] is the acceleration of gravity
t is the time of flight
Re-arranging and solving for t,
[tex]-9.5 = (12.5)(sin 45^{\circ}) t -\frac{1}{2}(9.8)t^2\\-9.5 = 8.8 t - 4.9 t^2\\4.9t^2-8.8t-9.5=0[/tex]
And solving for t,
t = -0.76 s
t = 2.55 s
We discard the first negative solution, so the balloon reaches the ground after 2.55 seconds.
B)
In order to find the horizontal distance covered by the balloon, we use the equation of the horizontal motion:
[tex]x=u_x t[/tex]
where:
[tex]u_x = u cos \theta[/tex] is the initial horizontal velocity, where
u = 12.5 m/s is the initial speed
[tex]\theta=45^{\circ}[/tex] is the angle of projection
t is the time
By substituting
t = 2.55 s (time of flight)
we find the horizontal distance covered by the balloon during this time:
[tex]x=(u cos \theta) t = (12.5)(cos 45^{\circ})(2.55)=22.5 m[/tex]
C)
In this case, the equation for the vertical motion would have been:
[tex]s=u_y t - \frac{1}{2}gt^2[/tex]
But this time,
s = 0
Because the vertical displacement is zero.
Therefore, the solution of the equation for t would be:
[tex]u_yt-\frac{1}{2}gt^2=0\\u_y - \frac{1}{2}gt=0\\t=\frac{2u_y}{g}=\frac{2(12.5)(sin 45^{\circ})}{9.8}=1.80 s[/tex]
And so, the horizontal distance covered by the balloon in this second case would be:
[tex]x=u_x t = (12.5)(cos 45^{\circ})(1.80)=15.9 m[/tex]
Learn more about projectile motion:
brainly.com/question/8751410
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