a canon is shot horizontally at 15 m/s off a cliff, which is 36m above water. how far away from the cliff should the target enemy boat be when the canon is shot?
the horizontal speed is constant.
So, how long does it take to fall 36 meters?
h(t) = 36 - 4.9t^2
As to the distance of the boat, it depends on the goal: hit or no hit?
The goal is to hit I believe.
To find the distance the target enemy boat should be from the cliff when the canon is shot, we need to consider the horizontal motion of the projectile.
Here's how to calculate it step by step:
1. Identify the known values:
- Initial velocity of the canon projectile (v₀) = 15 m/s
- Height of the cliff (h) = 36 m
- Acceleration due to gravity (g) = 9.8 m/s² (assuming no air resistance)
- Time of flight (t) - this is what we're trying to find
- Horizontal distance (d) - this is what we're trying to find
2. Analyze the vertical motion:
- The motion of the canon projectile in the vertical direction can be modeled using the equations of motion for free-falling objects.
- We can use the equation: h = v₀ * t - (1/2) * g * t²
- Rearranging the equation, we get: t² - (2v₀/g) * t - (2h/g) = 0
3. Solve for time (t):
- We can use the quadratic formula to solve for t in the equation t² - (2v₀/g) * t - (2h/g) = 0
- The formula is: t = (-b ± √(b² - 4ac)) / 2a
- Substituting the values into the formula, we get: t = (-(-2v₀/g) ± √((2v₀/g)² - 4(-2h/g))) / 2(1)
- Simplifying further: t = (2v₀/g ± √((2v₀/g)² + 8h/g)) / 2
- t = (2v₀/g ± √(4v₀²/g² + 8h/g)) / 2
- t = (v₀/g ± √(v₀²/g² + 2h/g))
4. Calculate time (t):
- Substituting the known values into the equation, we have: t = (15/9.8 ± √(15²/9.8² + 2*36/9.8))
- Simplifying further: t ≈ 1.5307 s
5. Calculate horizontal distance (d):
- The horizontal distance covered by the canon projectile is given by the equation: d = v₀ * t
- Substituting the values, we have: d = 15 * 1.5307
- Simplifying further: d ≈ 22.961 m
Therefore, the target enemy boat should be approximately 22.961 meters away from the cliff when the canon is shot.