A bullet with mass \(35.5 \mathrm{~g}\) is shot horizontally from a gun. The bullet embeds in a 5.90 -kg block of wood that is suspended by strings. The combined mass swings upward, gaining a height of \(12.85 \mathrm{~cm}\). What was the speed of the bullet as it left the gun? (Air resistance can be ignored here.)

Momentum, the product of an object's mass and velocity, has a fundamental role in physics. Conservation of momentum is a principle stating that the total momentum of a closed system of objects (meaning no external forces are acting on it) remains constant over time. During a collision or interaction between two bodies, like a bullet and a block of wood, their total momentum before the collision is equal to their total momentum after the collision. This principle simplifies calculations in systems where colliding objects stick together, like in our exercise, allowing us to solve for unknown velocities.

When we discuss motion and collisions, kinetic energy, the energy of an object in motion, is key. It's calculated using the formula \(\frac{1}{2}mv^2\), where \(m\) is mass and \(v\) is velocity. On the flip side, potential energy is the stored energy of an object due to its position or state. For example, when the bullet-block system in our exercise swings upwards, kinetic energy converts to gravitational potential energy, which is calculated with \(mgh\), where \(g\) is the acceleration due to gravity and \(h\) is the height. At the peak of its swing, all the system's kinetic energy has transformed into potential energy. This energy conservation allows us to relate velocity to the height the object reaches post-collision.

Projectile motion refers to the motion of an object thrown or projected into the air, subject only to the acceleration of gravity. The object is called a projectile, and its path is called its trajectory. The motion of a projectile can be broken into two components: horizontal and vertical. In ideal projectile motion, like the bullet in our textbook exercise, air resistance is typically ignored, leading to a constant horizontal velocity and a vertical acceleration due to gravity alone. As our problem's bullet-block system gained height, it essentially showcased projectile motion, where the vertical displacement was key to solving the problem.

Calculating the initial velocity is common in physics problems involving motion. The initial velocity is the velocity of an object before it undergoes acceleration or deceleration. To find it, we often use principles of conservation, like momentum and energy. In the context of our exercise, we sought the bullet's initial velocity. By equating the initial and final momentum and factoring in the energy conservation between kinetic and potential energy after the bullet embeds in the block, we find a formula involving masses, gravitational acceleration, and height. This formula lets us calculate the initial velocity before the impact, providing crucial insight into the bullet's speed as it left the gun.