A 1.00 kg block is attached to a horizontal spring with...

A $1.00 \mathrm{kg}$ block is attached to a horizontal spring with spring constant $2500 \mathrm{N} / \mathrm{m}$. The block is at rest on a friction less surface. A 10 g bullet is fired into the block, in the face opposite the spring, and sticks. a. What was the bullet's speed if the subsequent oscillations have an amplitude of $10.0 \mathrm{cm} ?$ b. Could you determine the bullet's speed by measuring the oscillation frequency? If so, how? If not, why not?

A $1.00 \mathrm{kg}$ block is attached to a horizontal spring with spring constant $2500 \mathrm{N} / \mathrm{m}$. The block is at rest on a friction less surface.
A 10 g bullet is fired into the block, in the face opposite the spring, and sticks.
a. What was the bullet's speed if the subsequent oscillations have an amplitude of $10.0 \mathrm{cm} ?$
b. Could you determine the bullet's speed by measuring the oscillation frequency? If so, how? If not, why not?

Key Concepts

Conservation of Momentum in Inelastic Collisions

In situations where two objects collide and stick together (a perfectly inelastic collision), momentum is conserved. This means that the total momentum before the collision is equal to the total momentum after the collision, even though kinetic energy is not conserved. This principle is vital for determining the velocity of the combined mass immediately after the collision, using the known masses and the initial speed of the moving object.

Conversion Between Kinetic Energy and Elastic Potential Energy

Once the collision has occurred, the kinetic energy of the moving combined mass is converted into elastic potential energy as the spring compresses and the oscillation reaches its amplitude. By equating the system's kinetic energy right after the collision to the elastic potential energy stored in the spring at maximum displacement, one can relate the initial kinetic properties to the amplitude of the resulting oscillation.

Simple Harmonic Motion Parameters

In a spring-mass system exhibiting simple harmonic motion, the oscillation frequency is determined solely by the mass and the spring constant. While the amplitude of the motion is indicative of the energy involved, the frequency does not depend on the energy delivered during the collision. This concept underscores why measuring the oscillation frequency alone does not provide sufficient information to determine the bullet's initial speed.

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