Assuming birdman flies at a height of 78m, how fast should birdman fly to hit the bucket if the bucket is placed 75m from the start of the field?

The speed of how fast the birdman should be to hit the bucket is 18.8 m/s

The only force acting on the height at which birdman flies is the gravitational force field.

Suppose the initial velocity (u) is parallel to the ground when the time (t) hits the ground.

Then, we can use the second equation of motion to determine the time (t) when the bird hits the ground.

Now, the speed of how fast the birdman should be to hit the bucket can be determined using;

Therefore, we can conclude that the speed of how fast the birdman should be to hit the bucket is 18.8 m/s

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I will suppose that:

The initial velocity of the birdman is horizontal.

Now, the only force acting on birdman will be the gravitational force, so we can write the acceleration of birdman as:

a(t)  = -9.8m/s^2 (the negative sign is because this force is pulling downwards)

To get the vertical velocity, we can integrate over time and get:

v(t) = (-9.8m/s^2)*t + V0

where t represents the time in seconds, and V0 is the initial vertical velocity, i already assumed that the initial velocity is only horizontal, so here V0  = 0m/s.

v(t) =  (-9.8m/s^2)*t

To find the vertical position as a function of time, we integrate again.

P(t) = (1/2)*(-9.8m/s^2)*t^2 + P0

Where P0 is the initial height, we know that it is 78m

Then the position is:

P(t) = (-4.9m/s^2)*t^2 + 78m.

Now, the bucket is placed in the ground, so he will reach the bucket when his vertical position is equal to zero, then we must solve:

P(t) = 0m =  (-4.9m/s^2)*t^2 + 78m.

(4.9m/s^2)*t^2  = 78m.

t = √(78m/(4.9m/s^2)) = 3.99 seconds.

Now, in the horizontal plane, he must travel 75 meters in a time of 3.99 seconds if he wants to hit the bucket.

Now we can use the equation

Distance = Speed*Time

75m = S*3.99s

S = 75m/3.99s = 18.8 m/s.

So the horizontal speed is 18.8 m/s.