Suppose a fixed cannon is to fire a projectile at an enemy tank, which is moving toward the cannon at a speed of 15 mph. If the cannon is to fire at the moment the tank is 3 miles from the cannon and the muzzle speed of the cannon is 1100 mph, what is the correct equation to determine the firing angle if acceleration is 79000 m /h^2?
Answer: (10.2cos(theta) + 0.139) sin(theta) = 1
To determine the firing angle, we need to consider the horizontal and vertical motion of the projectile. Let's break down the problem into components.
Horizontal Motion:
The horizontal motion of the projectile is unaffected by the acceleration or speed of the tank. Therefore, the time it takes for the projectile to reach the tank horizontally can be found using the formula:
time = distance / speed
time = 3 miles / 1100 mph
Vertical Motion:
The vertical motion of the projectile is influenced by gravity and the acceleration. We can use the equations of motion to determine the height h at time t:
h = ut + (1/2)at^2
where u is the initial vertical velocity, a is the acceleration, and t is time.
Given that the initial vertical velocity is 0 (since the projectile is fired horizontally), the equation can be simplified to:
h = (1/2)at^2
Since we need to find the firing angle, we can represent it as theta. The vertical displacement at the time the projectile reaches the tank would be the negative of this vertical displacement. Therefore:
h = -3 miles
Now, we can substitute the values in the equation:
(-3 miles) = (1/2)(79000 m/h^2) (time)^2
Since we have different units (miles and meters), we need to convert them to a common unit. Let's convert miles to meters:
1 mile = 1609.34 meters
Therefore:
(-3 miles) = (-3)(1609.34 meters) = -4827.34 meters
Now we can find the time it takes to reach the tank vertically using the previously obtained horizontal time. Let's substitute this time into the equation and solve for theta:
(-4827.34 meters) = (1/2)(79000 m/h^2)(time)^2
Solve for time by rearranging the equation:
(time)^2 = (-2)(-4827.34 meters) / (79000 m/h^2)
time = sqrt(-2)(-4827.34 meters) / (79000 m/h^2)
Now we have the time it takes for the projectile to reach the tank horizontally and vertically. With this information, we can find the firing angle theta using trigonometry.
sin(theta) = vertical displacement / horizontal displacement
sin(theta) = -3 miles / (1100 mph * time)
Since we have different units (miles and meters), we need to convert them to a common unit. Let's convert miles to meters and hours to seconds:
1 mile = 1609.34 meters
1 hour = 3600 seconds
Therefore:
sin(theta) = (-3)(1609.34 meters) / (1100 mph)(3600 seconds)
Simplify the equation:
sin(theta) = (-3)(1609.34 meters) / (1100 mph * 3600 seconds)
sin(theta) = -0.139 meters/second
Now we need to consider the horizontal component of the projectile's velocity. We can find the horizontal velocity using the formula:
velocity = distance / time
velocity = 3 miles / (time)
Substitute the previously obtained time:
velocity = 3 miles / (sqrt(-2)(-4827.34 meters) / (79000 m/h^2))
Convert miles to meters:
velocity = (3)(1609.34 meters) / (sqrt(-2)(-4827.34 meters) / (79000 m/h^2))
Simplify the equation:
velocity = 10.2 m/h^2
Now we can find the cosine of the firing angle since the horizontal velocity is given, and we already know the sine of the angle:
cos(theta) = horizontal velocity / muzzle speed
cos(theta) = 10.2 m/h^2 / 1100 mph
Since we have different units (meters and miles), we need to convert them to a common unit. Let's convert meters to miles:
1 meter = 0.000621371 miles
Therefore:
cos(theta) = (10.2)(0.000621371 miles) / 1100 mph
Simplify the equation:
cos(theta) = 0.000005956 miles/hour
The correct equation to determine the firing angle would be:
(10.2cos(theta) + 0.139)sin(theta) = 1