What is the maximum distance we can shoot a dart, provided our toy dart gun gives a maximum initial velocity of 5.84 m/s?

1) The maximum distance can be reached when the dart is shot with an angle of  above the horizontal (see demonstration of this fact at the end).

2) The motion of the dart is an uniform motion on the x-axis (horizontal direction) with constant velocity  and it is an uniformly accelerated motion on the y-axis (vertical direction), with the gravitational acceleration  acting downwards. So we can write the laws of motion on both directions:


where the negative sign means that g points downwards.

3) First of all we can find the time at which the dart reaches the ground. This can be found by requiring :

From this we find two solutions: , corresponding to the beginning of the motion (so we are not interested in this one), and 


4) Now that we now when the dart reaches the ground, we can use this information to find the distance covered on the x-axis, by using  inside the equation of  written at point 2:


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DEMONSTRATION OF POINT 1:

Calling v the initial velocity of the dart, and using a coordinate system where the x-axis coincides with the horizontal direction and the y-axis with the vertical direction, we can write the law of motion on both directions:


where t is the time, is the initial angle of the dart, and is the gravitational acceleration.
The maximum horizontal distance  can be found by requiring that . This condition occurs twice: when the motion starts () and when the dart falls to the ground (let’s call the time at which this happens). Therefore we can find by requiring , i.e.:

which has two solutions: (beginning of the motion) and .So we can find the maximum horizontal distance covered by the dart by substituting this into the law of motion for :
 
Since
,
we can write

Since g is negative, the maximum of this function occurs for , and this happens when .