The gun, mount, and train car of a railway had a total mass of 1.22 x 10^6 kg. The gun fired a projectile that was 80 cm in diameter and weighed 7502 kg. In the firing, the gun has been elevated 20 degrees above the horizontal. If the railway gun at rest before firing and moved to the right at a speed of 4.68 m/s immediately after firing, what was the speed of the projectile as it left the barrel (muzzle velocity)? How far will the projectile travel if air resistance is neglected? Assume that the wheel axles are frictionless.
Assuming the projectile is fired along the path of the railroad, conservation of momentum applies
before firing, the momentum is zero.
0 = final momentum=
masscarstuff*V+massprojectile*cos20*4.68
solve for V
How far?
a) time in air:
in the vertical,
vf=vi-9.8/2 * t^2
-4.68=4.68-4.9 t^2 solve for t
b) distance horizontal
d=vh*timeinair=4.68cos20*t
To solve this problem, we can use the principle of conservation of momentum and the principles of projectile motion. Let's break the problem down into steps:
Step 1: Calculate the total mass of the gun, mount, and train car.
Given:
Total mass of the gun, mount, and train car = 1.22 x 10^6 kg
Step 2: Calculate the momentum of the gun, mount, and train car before firing.
Momentum = mass x velocity
Given:
Initial velocity of the gun, mount, and train car = 0 m/s (at rest)
Using the formula, we find:
Momentum before firing = (1.22 x 10^6 kg) x (0 m/s) = 0 kg * m/s
Step 3: Calculate the momentum of the projectile after being fired.
Using the principle of conservation of momentum, the momentum of the gun, mount, and train car after firing will be equal to the momentum of the projectile.
Step 4: Calculate the angle of elevation in radians.
Given:
Elevation angle = 20 degrees
We need to convert this angle to radians using the conversion factor: radians = degrees x (pi/180)
Angle in radians = 20 degrees x (pi/180) ≈ 0.3491 radians
Step 5: Calculate the horizontal and vertical components of the projectile's velocity.
Given:
Diameter of the projectile = 80 cm = 0.8 m
Weight of the projectile = 7502 kg
Using the formula for weight, we can calculate the mass of the projectile:
Weight = mass x acceleration due to gravity (g)
7502 kg = mass x 9.8 m/s^2
mass = 7502 kg / 9.8 m/s^2 ≈ 765.1 kg
The horizontal component of the velocity can be calculated using the equation:
Velocity in the x-direction = initial velocity x cos(angle)
Given:
Initial velocity in the x-direction = unknown (let's call it Vx)
The vertical component of the velocity can be calculated using the equation:
Velocity in the y-direction = initial velocity x sin(angle)
Given:
Initial velocity in the y-direction = unknown (let's call it Vy)
Step 6: Calculate the momentum of the projectile.
Using the formula for momentum:
Momentum = mass x velocity
The mass of the projectile is 765.1 kg.
The total velocity of the projectile can be calculated using the Pythagorean theorem:
Total velocity = sqrt((Velocity in the x-direction)^2 + (Velocity in the y-direction)^2)
Step 7: Calculate the momentum of the gun, mount, and train car after firing.
Given:
Final velocity of the gun, mount, and train car in the x-direction = 4.68 m/s (to the right)
To calculate the final velocity of the gun, mount, and train car in the y-direction, we need to consider that there is no vertical acceleration after firing, so the final vertical velocity (Vfy) is equal to the initial vertical velocity (Vy).
Momentum after firing = (Mass of the projectile + Mass of the gun, mount, and train car) x (Total velocity of the projectile)
Step 8: Solve for Vx and Vy.
Using the known values from steps 5 and 7, solve for Vx and Vy.
Step 9: Calculate the muzzle velocity.
The muzzle velocity is the initial velocity of the projectile as it leaves the barrel. Since the projectile is fired horizontally (Vy = 0), the muzzle velocity is equal to the horizontal component of the projectile's velocity, Vx.
Step 10: Calculate the range of the projectile.
The range of the projectile is the horizontal distance it travels neglecting air resistance. To calculate this, we need to use the formula for range:
Range = (Muzzle velocity^2 x sin(2 x angle)) / gravity
Given:
Angle = 20 degrees
Gravity = 9.8 m/s^2
Using the known values, calculate the range of the projectile.
Following these steps, you should be able to find the muzzle velocity and the range of the projectile.