A ball is kicked horizontally at 8.0 m/s from a cliff 80 meters high. How far from the base of the cliff will the stone strike the ground?
32 meters
To find the distance from the base of the cliff where the ball will strike the ground, we can use the kinematic equations of motion.
First, let's identify the known values:
- Initial vertical velocity (vy) = 0 m/s (since the ball is kicked horizontally)
- Initial vertical displacement (Δy) = -80 m (negative because the ball is moving downwards)
- Acceleration due to gravity (g) = -9.8 m/s² (negative because it acts in the opposite direction of motion)
Using the kinematic equation: Δy = vy * t + (1/2) * g * t²
Since the ball is initially kicked horizontally, the time taken to fall vertically will be the same as if it were simply dropped from rest. We can solve for the time (t) it takes for the ball to fall to the ground.
Since the initial vertical velocity is 0, the equation becomes: Δy = (1/2) * g * t²
Plugging in the values, we get: -80 = (1/2) * (-9.8) * t²
Simplifying the equation: -80 = -4.9 * t²
Divide both sides by -4.9: t² = 80 / 4.9
Taking the square root of both sides: t ≈ √(80 / 4.9)
Calculating t gives us: t ≈ 4.04 seconds (rounded to two decimal places)
Now that we have the time it takes for the ball to strike the ground, we can determine the horizontal distance (d) covered by using the equation: d = vx * t
Since the ball is kicked horizontally, its initial horizontal velocity (vx) remains constant throughout its motion. Given that the horizontal velocity is 8.0 m/s, we can substitute in the values:
d = 8.0 m/s * 4.04 s
Calculating d gives us: d ≈ 32.32 meters (rounded to two decimal places)
Therefore, the stone will strike the ground approximately 32.32 meters away from the base of the cliff.
To find the horizontal distance the ball will travel before striking the ground, we can use the equation for horizontal motion:
distance = velocity * time
Since the ball is kicked horizontally, the initial vertical velocity is 0 m/s, and there is no acceleration in the horizontal direction. Thus, the time it takes for the ball to reach the ground will be the same as the time it takes for a free-falling object to fall from a height of 80 meters.
To find the time it takes for the ball to fall from a height of 80 meters, we can use the equation for vertical motion:
distance = 0.5 * acceleration * time^2
Since the initial velocity in the vertical direction is 0 m/s, the equation simplifies to:
80 = 0.5 * 9.8 * time^2
Solving for the time:
time^2 = 80 / (0.5 * 9.8)
time^2 = 16.33
Taking the square root of both sides:
time ≈ 4.04 seconds
Now, we can find the horizontal distance:
distance = velocity * time
distance = 8.0 * 4.04
distance ≈ 32.3 meters
Therefore, the ball will strike the ground approximately 32.3 meters from the base of the cliff.