A 2kg stone is ejected upwards at an angle of 35 degrees with an initial velocity of 8 m/s from a cannon with an unknown initial height and lands on top of a platform that is 3 meters tall. The horizontal distance from the cannon to the platform is 7 meters. What is the initial height of cannon from which the stone is launched?
the horizontal speed is 8 cos35° m/s, so how long does it take to cover the 7m?
The height h of the stone is
h(t) = h0 + (8 sin35°)t - 4.9t^2
plug in your t and solve for h0 when h(t) = 7
I mean h(t) = 3, the final height.
To find the initial height of the cannon from which the stone is launched, we can use the principles of projectile motion.
First, let's break down the given information:
- Mass of the stone (m): 2 kg
- Launch angle (θ): 35 degrees
- Initial velocity (v₀): 8 m/s
- Height of the platform (h): 3 meters
- Horizontal distance (d): 7 meters
Now, let's analyze the motion of the stone:
1. Resolve the initial velocity into horizontal and vertical components:
- Vertical component: v₀ * sin(θ)
- Horizontal component: v₀ * cos(θ)
2. Calculate the time for the stone to reach the highest point:
- Use the equation: v = u + at
Since the stone is ejected upwards, the final velocity at the highest point is 0 m/s. Thus, the equation becomes: 0 = v₀ * sin(θ) - g * tᵤ
where g is the acceleration due to gravity (-9.8 m/s²)
- Rearrange the equation to solve for tᵤ: tᵤ = v₀ * sin(θ) / g
3. Calculate the time of flight (total time):
- Since the stone goes up and then lands on the platform, the time of flight (T) is twice the time for reaching the highest point.
- Therefore, T = 2 * tᵤ
4. Find the horizontal distance covered during the time of flight:
- Use the equation: d = v₀ * cos(θ) * T
Plug in the known values: d = v₀ * cos(35°) * 2 * tᵤ
5. Calculate the vertical height reached by the stone:
- Use the equation: h = v₀ * sin(θ) * T + 0.5 * g * T²
Plug in the known values: h = v₀ * sin(35°) * 2 * tᵤ + 0.5 * (-9.8) * (2 * tᵤ)²
6. Determine the initial height of the cannon:
- Since the stone lands on top of the platform, the initial height of the cannon is equal to:
Initial height = height reached by the stone (h) + height of the platform (3 meters)
By following these steps and plugging in the given values, you can find the initial height of the cannon.