Final answer:
The initial horizontal velocity of the dart is approximately 26.63 m/s.
The correct option is not given.
Step-by-step explanation:
The horizontal motion of the dart is not affected by the vertical motion. The only force acting horizontally is the initial horizontal velocity of the dart. Therefore, the initial horizontal velocity (v_horizontal) can be found using the horizontal distance traveled by the dart.
The horizontal motion equation is given by:
Horizontal distance = v_horizontal x Time
Since the dart is thrown horizontally, there is no initial vertical velocity, and the time of flight can be determined using the vertical motion equation:
Vertical distance = {1} / {2} g t^2
In this case, the vertical distance (0.062 m) is the displacement below the target's center, and the acceleration due to gravity (g) is approximately (9.8 m/s^2).
Let's calculate the time of flight t using the vertical motion equation:
0.062 = {1} / {2} x 9.8 x t^2
Solving for t:
t^2 = {0.062 x 2} / {9.8}
t^2 ≈ 0.01265
t ≈ {0.01265}^ 1/2
t ≈ 0.1127 s
Now that we know the time of flight, we can use it to find the initial horizontal velocity using the horizontal motion equation:
Horizontal distance = v_horizontal x Time
3 m = v_horizontal x 0.1127 s
Solving for v_horizontal:
v_horizontal = {3} / {0.1127}
v_horizontal ≈ 26.63 m/s
Therefore, none of the provided options (A, B, C, D) match the calculated initial horizontal velocity. It's possible that there might be an error in the given options or in the information provided. Please double-check the problem or provide additional details.
The correct option is not given