A 500 g wood block on a frictionless table is attached to a horizontal spring. A 50 g dart is shot into the face of the block opposite the spring, where it sticks. Afterward, the spring oscillates with a period of 1.5 s and an amplitude of 20 cm. How fast was the dart moving when it hit the block?

Verified step by step guidance

1. Identify the initial and final masses involved in the collision. The initial mass of the dart is 50 g and the block is 500 g. After the collision, the dart sticks to the block, so the combined mass is 550 g.
2. Understand that the collision is perfectly inelastic since the dart sticks to the block. Use the conservation of momentum principle, which states that the total momentum before the collision must equal the total momentum after the collision.
3. Set up the equation for conservation of momentum. Let $v$ be the initial velocity of the dart, and $u$ be the velocity of the block and dart immediately after the collision. The equation is: $0.05 \text{ kg} \cdot v + 0.5 \text{ kg} \cdot 0 = 0.55 \text{ kg} \cdot u$.
4. Determine the relationship between the amplitude, period, and velocity of the oscillating system. The maximum velocity $u$ of the block and dart can be found from the amplitude and angular frequency of the spring. The angular frequency $\omega$ can be calculated using the period $T = 1.5 \text{ s}$, where $\omega = \frac{2\pi}{T}$.
5. Use the relationship $u = A \omega$ to find the velocity $u$ immediately after the collision, where $A = 0.2 \text{ m}$ is the amplitude. Substitute $u$ back into the momentum conservation equation to solve for $v$, the initial velocity of the dart.

Key Concepts

Conservation of Momentum

The principle of conservation of momentum states that in a closed system, the total momentum before an event must equal the total momentum after the event, provided no external forces act on it. In this scenario, when the dart collides with the wood block, the combined momentum of the dart and block system must be conserved, allowing us to calculate the dart's initial velocity based on the final velocity of the block-dart system.

Simple Harmonic Motion (SHM)

Simple Harmonic Motion refers to the oscillatory motion of an object where the restoring force is directly proportional to the displacement from its equilibrium position. In this case, the spring attached to the block exhibits SHM, characterized by its period and amplitude, which are essential for understanding the dynamics of the system after the dart impacts the block.

Spring Constant and Period of Oscillation

The period of oscillation of a mass-spring system is determined by the mass attached to the spring and the spring constant, which measures the stiffness of the spring. The formula T = 2π√(m/k) relates the period (T) to the mass (m) and spring constant (k). Knowing the period allows us to derive the spring constant, which can be used to analyze the energy transfer during the collision and subsequent motion.