Question

An injured monkey sits perched on a tree branch 6.0 m above the ground, while a wildlife veterinarian is kneeling down in the bushes 86.5 m away attempting to subdue the monkey with a tranquilizer gun. The vet knows that the moment the gun fires the monkey will be frightened and fall down from the branch. What angle up from the ground must the veterinarian aim the gun so that the tranquilizer dart will hit the falling monkey?

Answer 1

Let´s stablish

+y = vertical, upwards. y=0 corresponds to the ground level.

+x= horizontal, towards the monkey. x=0 corresponds to the firing position of the dart.

Then, the dart and the monkey should be at the same position in y in a time t seconds.

The position of the dart can be described as

  where g is the acceleration of gravity, and t is the elapsed time. V_y is the component of the firing velocity in the y direction.

Also, the position of the monky is given by

where y₀ is the initial position of the monkey, y₀= 6.0 m.

The dart will hit the monkey when those positions are the same, then,

If the gun is fired at a positive non-zero angle from the ground, we should have a non-zero value for V_y, then , we can write

On the other hand, the dart needs to cover 86.5 m horizontally, in t seconds to hit the monkey, so

Finally, the angle at which the gun should be fired is given by