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
I just need help coming up with that answer
You can get your acceleration equation a=<0, -79000> and get your position equation by integrating and using your initial values so you'll get
s(t) = <1100cos(theta)*t, -39500*t^2 + 1100sin(theta)*t>
then set your y component equal to zero to find how long it will take the ball to hit the "ground"(which will be the tank).
Once you have the "t" with respect to theta, you say the tank moves towards you at 15mph and is initially 3 miles away.
so.. its x-position at time "t" will be 3-15*t(where t is the time you found earlier)
You then set that x-position of the tank to the x-position of the cannonball and figure out what theta is needed.
It will most likely give you an equation that looks different than that equation but doing some simple algebra will give you that.
To determine the firing angle, we can use the principles of projectile motion. The key idea is that both the tank and the projectile have horizontal and vertical components of motion.
First, let's establish the given values:
- The muzzle speed of the cannon is 1100 mph.
- The tank is moving toward the cannon at a speed of 15 mph.
- The tank is 3 miles away from the cannon when the cannon fires.
- The acceleration is 79000 m/h^2.
Now, let's break down the problem step by step:
Step 1: Convert all the given values to consistent units.
- Convert the muzzle speed of the cannon from mph to m/h: 1100 mph * 1.60934 km/m * 1000 m/km / 1 h = 1770240 m/h.
- Convert the speed of the tank from mph to m/h: 15 mph * 1.60934 km/m * 1000 m/km / 1 h = 24140 m/h.
- Convert the distance from miles to meters: 3 miles * 1609.34 m/mile = 4828.02 m.
Step 2: Determine the time it takes for the projectile to reach the tank.
Since the tank is moving towards the cannon, the relative speed between the cannon and the tank is the sum of their speeds: relative speed = muzzle speed - tank speed = 1770240 m/h - 24140 m/h = 1746100 m/h.
Using the equation distance = speed * time, we can rearrange it to time = distance / speed.
Thus, the time it takes for the projectile to reach the tank is t = 4828.02 m / 1746100 m/h = 0.002765 h.
Step 3: Calculate the vertical displacement of the projectile during this time.
Since the initial vertical velocity is 0 (assuming the cannon is fired horizontally), we can use the equation of motion, s = ut + (1/2)at^2, where:
- u is the initial vertical velocity (0),
- a is the constant acceleration (-79000 m/h^2),
- t is the time (0.002765 h),
- s is the vertical displacement we want to calculate.
Substituting the values, we have 0 = 0 + (1/2)(-79000 m/h^2)(0.002765 h)^2 + s.
Simplifying, we have s = 0.0714 m.
Step 4: Calculate the horizontal displacement of the projectile during this time.
Since the initial horizontal velocity is the muzzle speed of the cannon (1770240 m/h), we can use the equation distance = speed * time, where:
- distance is the horizontal displacement we want to calculate,
- speed is the initial horizontal velocity (1770240 m/h),
- time is the time (0.002765 h).
Substituting the values, we have distance = 1770240 m/h * 0.002765 h = 4896.16 m.
Step 5: Calculate the angle.
We can use the trigonometric relationship tangent(theta) = vertical displacement / horizontal displacement, where:
- theta is the firing angle we want to calculate,
- vertical displacement is 0.0714 m,
- horizontal displacement is 4896.16 m.
Substituting the values, we have tan(theta) = 0.0714 m / 4896.16 m.
Taking the inverse tangent of both sides, we have theta = arctan(0.0714 m / 4896.16 m).
Step 6: Convert the angle to degrees.
The result from Step 5 is in radians, so we need to convert it to degrees.
Using the conversion formula, we have theta (in degrees) = theta (in radians) * (180 degrees / pi radians).
Substituting the values, we have theta (in degrees) = arctan(0.0714 m / 4896.16 m) * (180 degrees / pi radians) = 0.087 degrees.
Therefore, the correct equation to determine the firing angle, when considering the given values, would be:
tan(theta) = 0.0714 m / 4896.16 m.