Final answer:
The impact speed of the projectile launched horizontally off a 78-meter cliff with a velocity of 27 m/s will be 17 m/s.
Step-by-step explanation:
To find the impact speed of the projectile, we need to consider the vertical motion and the horizontal motion separately. Since the projectile is launched horizontally, it will have a constant horizontal velocity of 27 m/s throughout its motion. The vertical motion can be analyzed using the equation:
Δy = V0yt + (1/2)gt2
Where Δy is the change in vertical position, V0y is the initial vertical velocity (which is 0 in this case since it is launched horizontally), g is the acceleration due to gravity (-10 m/s²), and t is the time of flight.
Since the projectile is launched horizontally, it will take the same amount of time to reach the ground as if it were dropped from the same height (78 meters) vertically. Using the formula t = √(2h/g), we find that t = √(2*78/10) = 8.85 seconds. Substituting this value into the first equation, we get:
78 = 0 + (1/2)(-10)(8.85)2
78 = -39.15t2
Solving for t, we find that t = 1.00 second. This is the time it takes for the projectile to hit the ground. Now we can calculate the impact speed using the formula:
V = V0 + gt
Substituting the given values, we get:
V = 27 + (-10)(1.00)
V = 17 m/s
Therefore, the impact speed of the projectile will be 17 m/s.