Final answer:
The dart will hit the ground approximately 2.63 seconds after being thrown down from a 75.0 m cliff at an initial velocity of 10.0 m/s.
Step-by-step explanation:
To find the time it takes for the dart to hit the ground, we can use the equation: h = v0t + (1/2)gt2, where h is the height, v0 is the initial velocity, g is the acceleration due to gravity, and t is the time.
Since the dart is thrown downwards, the initial velocity will be negative (-10.0 m/s), and the acceleration due to gravity is -9.8 m/s2.
Plugging in the values, we have: 75.0 = -10.0t + (1/2)(-9.8)t2. Rearranging the equation, we get a quadratic equation: 4.9t2 - 10.0t + 75.0 = 0.
Solving this quadratic equation gives us two solutions, but we can ignore the negative solution as time cannot be negative. So, the time it takes for the dart to hit the ground is approximately 2.63 seconds.