An antiaircraft gun fires at an elevation of 60° at an enemy aircraft at 10,000m above the ground. At what speed must the cannon be shot to hit the plane at that height.
Y^2 = Yo^2 + 2g*h = 0
Yo^2 = -2g*h = -2*(-9.8)*10,000 = 196,000
Yo = 443 m/s.=Ver. component of initial
velocity.
Vo = Yo/sin60 = 443/sin60 = 512 m/s =
Initial velocity = Speed at which the
cannon is fired.
To calculate the required speed of the cannonball to hit the aircraft at a certain height, we can use the principles of projectile motion. In this scenario, the cannonball can be considered as a projectile and follows a curved trajectory.
First, let's break down the information we have:
- Angle of elevation (θ): 60°
- Height of the enemy aircraft (h): 10,000m
The key variables we need to determine are the initial velocity (v) of the cannonball and the time of flight (t) of the projectile.
To calculate the initial velocity (v), we can use the horizontal and vertical components of the velocity separately. Let's assume the initial velocity is v₀, and it can be separated into its horizontal (v₀x) and vertical (v₀y) components.
v₀x = v₀ * cos(θ)
v₀y = v₀ * sin(θ)
Next, we'll calculate the time of flight (t). Since the projectile follows a parabolic path, the time of flight can be determined using the vertical component of the motion. We'll use the equation:
y = v₀y * t - (1/2) * g * t²
where:
- y is the height of the aircraft (10,000m)
- v₀y is the vertical component of the initial velocity
- g is the acceleration due to gravity (approximately 9.8 m/s²)
- t is the time of flight
Since the projectile reaches its maximum height when the vertical velocity becomes zero (at the peak of the trajectory), we can use this fact to determine the time of flight. At the peak, v₀y = 0, so we can solve the equation above for t.
0 = v₀y * t - (1/2) * g * t²
This equation is a quadratic equation in terms of t, and we can solve it using the quadratic formula.
By substituting the respective values into the equation and solving for t, we can find the time of flight.
Once we have the time of flight, we can determine the horizontal distance traveled (x) by the projectile:
x = v₀x * t
Now, let's calculate the required speed:
1. Calculate the vertical component of the initial velocity (v₀y):
v₀y = v₀ * sin(θ)
2. Calculate the time of flight (t):
0 = v₀y * t - (1/2) * g * t²
Solve for t using the quadratic formula.
3. Calculate the horizontal component of the initial velocity (v₀x):
v₀x = v₀ * cos(θ)
4. Calculate the horizontal distance traveled (x):
x = v₀x * t
To hit the plane at the given height, the speed at which the cannonball must be shot is equal to the magnitude of the initial velocity (v). Hence, v = √(v₀x² + v₀y²).
To find the speed at which the cannon must be shot to hit the plane at that height, we can break down the problem into components and use trigonometry.
Let's consider the horizontal and vertical components of the cannon's velocity. The horizontal component of the velocity will remain constant throughout the flight, while the vertical component will change due to gravity.
Given:
Elevation angle (θ) = 60°
Height of the plane (h) = 10,000m
Acceleration due to gravity (g) = 9.8 m/s²
We can start by finding the time taken for the projectile to reach the height of the plane using the vertical component of velocity.
Vertical component of velocity (v₀) = v₀ * sin(θ)
Time taken to reach the height of the plane = t = h / (v₀ * sin(θ))
Next, we can find the horizontal component of velocity using the time obtained above.
Horizontal component of velocity (v) = v₀ * cos(θ)
Horizontal distance traveled = d = v * t
Since the horizontal distance traveled is equal to the range of the projectile, we can substitute d in terms of the range (R):
R = v * t
Finally, solving for the velocity (v):
v = R / t
Let's calculate the velocity using the given values.
Step 1: Calculate the time taken to reach the height of the plane:
t = h / (v₀ * sin(θ))
= 10,000 / (v₀ * sin(60°))
Step 2: Calculate the horizontal component of velocity:
v = v₀ * cos(θ)
= v₀ * cos(60°)
Step 3: Calculate the velocity required to hit the plane:
v = R / t
= R / (10,000 / (v₀ * sin(60°)))
Since we don't know the range (R), we cannot calculate the exact value of the velocity without additional information. However, using this method, we can find the relationship between the range, angle, and velocity.