To find how far from the base of the cliff the stone will strike the ground, we need to calculate the time it takes for the stone to hit the ground and then use this time to find the horizontal distance it travels.

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Step 1: Calculate the time of flight

The vertical motion of the stone is influenced only by gravity. The vertical distance $h$ is given as 80 m, and the acceleration due to gravity $g$ is $9.8 \, \text{m/s}^2$.

Using the formula for vertical displacement under constant acceleration: $h = \frac{1}{2} g t^2$ Rearranging to solve for $t$: $t = \sqrt{\frac{2h}{g}}$ Substitute $h = 80 \, \text{m}$ and $g = 9.8 \, \text{m/s}^2$: $t = \sqrt{\frac{2(80)}{9.8}} = \sqrt{\frac{160}{9.8}} \approx \sqrt{16.33} \approx 4.04 \, \text{s}$

So, the time of flight is approximately $t = 4.04 \, \text{s}$.

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Step 2: Calculate the horizontal distance

The horizontal motion has a constant velocity $v_x = 8.0 \, \text{m/s}$. The horizontal distance $d$ is given by: $d = v_x t$ Substitute $v_x = 8.0 \, \text{m/s}$ and $t = 4.04 \, \text{s}$: $d = 8.0 \times 4.04 = 32.32 \, \text{m}$

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Final Answer:

The stone will strike the ground approximately 32.3 m from the base of the cliff.

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Would you like me to provide further explanation or additional variations of the problem? Here are some related questions:

1. What would happen if the initial velocity were doubled?
2. How does air resistance affect the motion of the stone?
3. What is the total velocity of the stone when it hits the ground?
4. How can we calculate the angle of impact with the ground?
5. What if the stone were thrown upward or downward instead of horizontally?

Tip: Always break projectile motion problems into vertical and horizontal components for easier analysis.