To determine if Anton will hit the bull's eye, we need to analyze the projectile motion of the dart.
First, let's consider the horizontal motion:
Anton throws the dart horizontally with a speed of 20 meters per second. Since there is no vertical force acting on the dart horizontally, the dart will continue to move with a constant horizontal velocity throughout its motion. Therefore, the horizontal distance the dart will travel is solely dependent on the time of flight.
Next, let's look at the vertical motion:
Anton throws the dart from the same height as the top of the bull's eye. Assuming the dart is subject to the force of gravity only, it will follow a parabolic path called a projectile motion. During this motion, the dart will experience both horizontal and vertical displacement.
To determine if Anton will hit the bull's eye, we need to calculate whether the horizontal distance covered by the dart in the time of flight will be within the 5 centimeters (0.05 meters) width of the bull's eye.
To do that, let's calculate the time of flight of the dart:
Using the equation of motion for vertical displacement, we can calculate the time it takes for the dart to reach the ground. Since the dart is thrown horizontally, the initial vertical velocity is zero.
Vertical displacement (h) = 0 (since the height from which the dart is thrown is the same as the height of the bull's eye)
Initial vertical velocity (vâ‚€) = 0
Acceleration due to gravity (g) = 9.8 m/s² (assuming no air resistance)
Using the formula:
h = v₀ * t + (1/2) * g * t²
0 = 0 * t + (1/2) * 9.8 * t²
Simplifying the equation, we have:
4.9t² = 0
Since the term is zero, we know the time of flight is zero. This means that the dart will hit the ground instantly.
Since the dart hits the ground instantly, it does not have enough time to travel horizontally. Therefore, Anton will not hit the bull's eye.