anton plan that he can throw at a dart board from a distance of 2 meters and hit the 5 centimeters wide bull's eye if he throw the dart horizontally with a speed of 20 meters per second. he start the throw at the height as the top of bull's eye ,will anton hit the bull's eye?
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 (v₀) = 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.
To determine whether Anton will hit the bull's eye, we need to analyze the projectile motion of the dart.
First, let's calculate the time it would take for the dart to travel the horizontal distance of 2 meters. We can use the equation:
distance = speed × time
Rearranging the equation:
time = distance / speed
time = 2 meters / 20 meters per second
time = 0.1 seconds
Now, let's determine the vertical distance the dart would fall in 0.1 seconds. We can use the equation of motion:
vertical distance = (1/2) × acceleration × time^2
Since the dart is thrown horizontally, there is no vertical acceleration. Therefore, the vertical distance would be zero, meaning the dart would not fall in the 0.1 seconds.
Considering that the throw starts at the same height as the top of the bull's eye, and the dart does not fall vertically during its travel time, Anton will hit the bull's eye.