To find the recoil velocity of the rifle, we can apply the principle of conservation of momentum. Since no external forces are acting on the system (rifle + bullet), the total momentum before and after firing must be conserved.

Step-by-Step Solution:

1. Initial Momentum:
   Before the rifle is fired, both the rifle and the bullet are at rest, so the total initial momentum is zero.

   $\text{Initial Momentum} = 0$

2. Final Momentum:
   After firing, the momentum of the bullet is in one direction, and the momentum of the rifle (recoil) is in the opposite direction.

   Let:

   • $m_r$ be the mass of the rifle = 1.70 kg
   • $v_r$ be the recoil velocity of the rifle (which we are solving for)
   • $m_b$ be the mass of the bullet = 0.0080 kg
   • $v_b$ be the velocity of the bullet = 770 m/s

   The final momentum is: $\text{Final Momentum} = (m_r \cdot v_r) + (m_b \cdot v_b) = 0$

   Since the bullet and rifle move in opposite directions, we can write: $m_r \cdot v_r = - m_b \cdot v_b$

3. Solving for Recoil Velocity $v_r$: $v_r = -\frac{m_b \cdot v_b}{m_r}$

   Substituting the values: $v_r = -\frac{(0.0080 \, \text{kg}) \cdot (770 \, \text{m/s})}{1.70 \, \text{kg}}$

   $v_r = -\frac{6.16}{1.70} \, \text{m/s}$

   $v_r = -3.62 \, \text{m/s}$

The negative sign indicates that the rifle moves in the opposite direction of the bullet.

Final Answer:

The recoil velocity of the rifle is 3.62 m/s (in the opposite direction of the bullet).

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Would you like more details on this calculation, or have any questions? Here are 5 related questions to further explore:

1. How does the mass of the bullet affect the recoil velocity of the rifle?
2. What would happen to the recoil velocity if the mass of the rifle increased?
3. How would the inclusion of the person holding the rifle affect the calculation?
4. Can we calculate the kinetic energy of the bullet and rifle after the shot?
5. How does conservation of momentum apply in real-world shooting scenarios with external forces?

Tip: Always consider direction when dealing with momentum — velocity is a vector quantity!