A military gun is mounted on a railroad car (m = 1590 kg) as shown in Figure P7.34. There is no frictional force on the car, the track is horizontal, and the car is initially at rest. The gun then fires a shell of mass 26 kg with a velocity of 280 m/s at an angle of 28° with respect to the horizontal. Find the recoil velocity of the car.
massshell*velocityshellhoriz+massgun*velocitygun=0
Vcar= -Vshell*cos28*massshell/masscar
To find the recoil velocity of the car, we can use the principle of conservation of momentum. According to this principle, the initial momentum of the system (car and gun) must be equal to the final momentum of the system.
We can break down the problem into two components: the horizontal and vertical direction.
In the horizontal direction, there is no external force acting on the system, except for the internal forces between the gun and the car. Since there is no friction, these internal forces do not affect the motion of the system. Therefore, the horizontal component of the momentum is conserved.
In the vertical direction, there is a vertical component of the momentum due to the shell being fired at an angle. However, this vertical component does not affect the horizontal motion of the car. The only relevant component is the horizontal component of the velocity of the shell.
Let's calculate the horizontal component of the velocity of the shell:
Vx = V * cos(θ)
= 280 m/s * cos(28°)
≈ 248.98 m/s (rounding to two decimal places)
Now, we can find the initial and final momenta in the horizontal direction. Since the car is initially at rest, its initial momentum is zero. The final momentum of the system must be zero to conserve momentum.
Let Vc be the recoil velocity of the car. The mass of the shell is 26 kg, and the mass of the car is 1590 kg.
Initial momentum: 0
Final momentum: (m * Vc) + (m * Vx) = 0
1590 kg * Vc + 26 kg * 248.98 m/s = 0
Solving the equation for Vc:
1590 kg * Vc = -26 kg * 248.98 m/s
Vc = (-26 kg * 248.98 m/s) / 1590 kg
Vc ≈ -4.0378 m/s (rounding to four decimal places)
The negative sign indicates that the recoil velocity is directed opposite to the initial direction of the shell. Therefore, the recoil velocity of the car is approximately -4.0378 m/s.