A 1100-kg cannon shoots a 31-kg shell at an angle of 25¡ above the horizontal and a speed of 520 m/s. What is the recoil velocity of the cannon? (Take the direction the cannon is firing to be the positive direction. Indicate the direction with the sign of your answer.)

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This problem can be solved by assuming conservation of horizontal momemtum

To calculate the recoil velocity of the cannon, we need to apply the principle of conservation of momentum. According to this principle, the total momentum before the firing of the cannon must equal the total momentum after.

The initial momentum of the system (cannon + shell) is zero since both the cannon and shell are at rest initially. Therefore, the total momentum after the cannon is fired (recoil velocity) will also be zero.

Let's break down the problem into components using vectors. The horizontal direction will be considered the x-axis, and the vertical direction will be considered the y-axis.

First, let's find the initial momentum of the shell before it is fired:

m_shell = 31 kg (mass of the shell)
v_shell = 520 m/s (initial velocity of the shell)

The initial momentum of the shell is given by:

P_shell = m_shell * v_shell

P_shell = 31 kg * 520 m/s

Next, let's find the vertical component of the shell's initial momentum before it is fired:

P_y_shell = P_shell * sin(25°)

Next, let's find the horizontal component of the shell's initial momentum before it is fired:

P_x_shell = P_shell * cos(25°)

According to the principle of conservation of momentum, the total momentum after the cannon is fired (recoil velocity) must be zero. Let's call the recoil velocity of the cannon V_cannon.

The total momentum after the cannon is fired will be the sum of the recoil velocity of the cannon and the momentum of the shell:

P_total = m_cannon * V_cannon + P_x_shell

Since the total momentum is zero, we can set the equation to:

0 = m_cannon * V_cannon + P_x_shell

Finally, we can solve for V_cannon:

V_cannon = -P_x_shell / m_cannon

Now that we've explained the steps, let's calculate the recoil velocity of the cannon using the given values:

m_cannon = 1100 kg (mass of the cannon)
P_x_shell = P_shell * cos(25°)

Using the values, we can calculate:

P_x_shell = P_shell * cos(25°) = (31 kg * 520 m/s) * cos(25°)

Now, we can substitute the values into the formula to find the recoil velocity:

V_cannon = -P_x_shell / m_cannon

After evaluating the expression, you will find the recoil velocity of the cannon.