An anti aircraft gun fires at an elevation of 60degree at an enemy aircraft at 10000m above the ground.at what speed must the cannon be shot to hit the plane at that height?(take g = 9.8m/s.)
m g h < 1/2 m [v sin(60º)]^2
v sin(60º) > √(2 g h)
Range = Vo^2*sin(2A)/g = 10,000.
Vo^2*sin(120)/9.8 = 10,000,
Vo^2*0.0884 = 10,000,
Vo = 336.4 m/s.
correct
To find the speed at which the cannon must be shot to hit the plane at a height of 10,000m, we can break the problem down into horizontal and vertical components.
First, let's consider the horizontal component.
The horizontal velocity remains constant throughout the motion because there are no horizontal forces acting on the projectile. Therefore, the horizontal component of the velocity can be calculated using the formula:
Horizontal velocity (Vx) = Initial velocity (Vo) × cos(angle)
In this case, the angle of elevation is given as 60 degrees. We need to convert this angle from degrees to radians to use it in the equation.
So, angle in radians = angle in degrees × (π/180)
Angle in radians = 60 × (π/180) ≈ 1.0472 radians
Now we can calculate the horizontal component of the velocity:
Vx = Vo × cos(1.0472)
Next, let's consider the vertical component.
The vertical motion of the projectile is affected by gravity. The equation to calculate the time of flight (t) is as follows:
Time of flight (t) = (2 × Vy) / g
Where Vy is the vertical component of the velocity and g is the acceleration due to gravity (9.8 m/s²).
Since the time of flight is the same for both the horizontal and vertical motions, we can substitute t into the equation for horizontal motion:
Vx = (Vo × cos(1.0472)) × t
Now we need to find the time of flight using the vertical motion.
The vertical displacement (or height) is given as 10,000m, and we want to find the vertical component of the velocity (Vy). We can use the equation for vertical displacement:
Vertical displacement = (Vy × t) - (1/2) × g × t²
Plugging in the values, we get:
10,000 = (Vy × t) - (1/2) × 9.8 × t²
We know that time of flight is the same as the time calculated earlier:
t = (2 × Vy) / g
By substituting this value of t into the equation for vertical displacement, we can solve for Vy:
10,000 = (Vy × [(2 × Vy) / g]) - (1/2) × 9.8 × [(2 × Vy) / g]²
Simplifying the equation will give us a quadratic equation in terms of Vy. We can solve it using the quadratic formula or by graphing the equation and finding the value of Vy when the height reaches 10,000m.
Once we have the value of Vy, we can substitute it back into the equation for horizontal motion:
Vx = (Vo × cos(1.0472)) × [(2 × Vy) / g]
Finally, we can calculate the initial velocity (Vo) using the Pythagorean theorem:
Vo = √(Vx² + Vy²)
These calculations will give us the speed at which the cannon must be shot to hit the plane at a height of 10,000m.