BIO Squids rely on jet propulsion to move around. A 1.50 kg squid (including the mass of water inside the squid) drifting at 0.40m/s suddenly ejects 0.100 kg of water to get itself moving at 2.50m/s . If drag is ignored over the small interval of time needed to expel the water (the impulse approximation), what is the water’s ejection speed relative to the squid?

Verified step by step guidance

1. Start by identifying the principle of conservation of momentum. Since no external forces act on the system (squid + water), the total momentum before and after the ejection must remain constant.
2. Write the equation for the total momentum before the ejection: \( p_{\text{initial}} = m_{\text{squid}} v_{\text{squid, initial}} + m_{\text{water}} v_{\text{water, initial}} \). Here, \( v_{\text{water, initial}} \) is the same as \( v_{\text{squid, initial}} \) because the water is initially inside the squid and moving with it.
3. Write the equation for the total momentum after the ejection: \( p_{\text{final}} = m_{\text{squid}} v_{\text{squid, final}} + m_{\text{water}} v_{\text{water, final}} \). Here, \( v_{\text{water, final}} \) is the velocity of the ejected water relative to the ground.
4. Set the total initial momentum equal to the total final momentum: \( m_{\text{squid}} v_{\text{squid, initial}} + m_{\text{water}} v_{\text{water, initial}} = m_{\text{squid}} v_{\text{squid, final}} + m_{\text{water}} v_{\text{water, final}} \). Substitute the known values for masses and velocities, leaving \( v_{\text{water, final}} \) as the unknown.
5. To find the water's ejection speed relative to the squid, use the relationship \( v_{\text{ejection}} = v_{\text{water, final}} - v_{\text{squid, final}} \). Solve for \( v_{\text{ejection}} \) using the value of \( v_{\text{water, final}} \) obtained from the momentum equation.

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, the squid and the expelled water form a closed system, allowing us to apply this principle to find the water's ejection speed.

Impulse and Momentum Change

Impulse is defined as the change in momentum resulting from a force applied over a period of time. In the context of the squid's movement, the impulse generated by the ejection of water results in a change in the squid's momentum, which can be calculated to determine the speed of the expelled water relative to the squid.

Relative Velocity

Relative velocity refers to the velocity of one object as observed from another object. In this problem, we need to calculate the speed of the water relative to the squid, which involves considering both the squid's final speed and the initial conditions before the water is expelled.