Final answer:
The question involves calculating the range and final vertical velocity of a dart launched horizontally from a height, using principles of projectile motion in Physics. The horizontal range is calculated by multiplying the time in the air by the horizontal velocity, and the final vertical velocity is found by multiplying gravitational acceleration by the time spent in the air.
Step-by-step explanation:
The subject of this question involves calculations related to projectile motion, which falls under the branch of Physics. The problem describes a dart that's launched horizontally from a height with no initial vertical velocity, which is a common scenario in projectile motion problems. To find the range and final vertical velocity of the dart, we can use kinematic equations and consider the motion separately in vertical and horizontal directions. The horizontal motion will be affected only by the initial horizontal velocity, while the vertical motion will involve gravitational acceleration.
For the horizontal range, the equation d = vt can be used, where d is the range, v is the horizontal velocity, and t is the time in air. Since the initial vertical velocity is zero, the time t the dart spends in the air can be found using t = √(2h/g), where h is the height and g is the acceleration due to gravity (9.81 m/s²). Once t is known, multiply it by the horizontal velocity to get the range.
The final vertical velocity can be found using the equation vf = gt, where vf is the final vertical velocity. Since there's no initial vertical velocity and the acceleration due to gravity is constant (downwards), the dart will gain speed in the vertical direction until it hits the ground. By substituting t from the previous calculation and g, we can find vf.