Problem 19 - University Physics with Modern Physics

Question

Animal Propulsion. Squids and octopuses propel themselves by expelling water. They do this by keeping water in a cavity and then suddenly contracting the cavity to force out the water through an opening. A \(6.50-\) kg squid (including the water in the cavity) at rest suddenly sees a dangerous predator. (a) If the squid has 1.75 \(\mathrm{kg}\) of water in its cavity, at what speed must it expel this water to suddenly achieve a speed of 2.50 \(\mathrm{m} / \mathrm{s}\) to escape the predator? Neglect any drag effects of the surrounding water. (b) How much kinetic energy does the squid create by this maneuver?

Step-by-Step Solution

Answer

(a) 6.79 m/s, (b) 14.84 J

Step 1: Identify essential information

The total mass of the squid including water is 6.50 kg. Out of this, the water mass is 1.75 kg, leaving the squid's own mass as 4.75 kg (6.50 kg - 1.75 kg). The desired final speed for the squid is 2.50 m/s. We assume conservation of momentum between the squid and the expelled water, as there is no external drag.

Step 2: Apply the conservation of momentum

The conservation of momentum states that the total initial momentum of the system (squid + water) should equal the total final momentum of the system. Initially, both are at rest, so:\[m_\text{squid} \cdot v_\text{squid final} + m_\text{water} \cdot v_\text{water final} = 0\]Rearranging gives us:\[m_\text{squid} \cdot 2.50 \, \text{m/s} = m_\text{water} \cdot v_\text{water final}\]

Step 3: Substitute known values into the momentum equation

Substituting the known values gives:\[4.75 \, \text{kg} \cdot 2.50 \, \text{m/s} = 1.75 \, \text{kg} \cdot v_\text{water final}\]Solving for \(v_\text{water final}\), we get:\[v_\text{water final} = \frac{4.75 \, \text{kg} \cdot 2.50 \, \text{m/s}}{1.75 \, \text{kg}} = 6.79 \, \text{m/s}\]

Step 4: Calculate the kinetic energy of the squid

The kinetic energy (KE) of the squid is given by the formula:\[\text{KE} = \frac{1}{2} m_\text{squid} \cdot v_\text{squid final}^2\]Substituting the known values, we have:\[\text{KE} = \frac{1}{2} \cdot 4.75 \, \text{kg} \cdot (2.50 \, \text{m/s})^2 = 14.84 \, \text{J}\]

Key Concepts

Kinetic Energy

Kinetic energy is the energy of motion. Whenever an object moves, it possesses kinetic energy. The amount of kinetic energy depends on two main factors: the mass of the object and its velocity. For example, in the case of the squid, part of its energy when escaping a predator is kinetic.

To calculate kinetic energy, we use the formula:

• KE = \(\frac{1}{2} m v^2\)

where \(m\) is the mass of the object and \(v\) is its velocity.

In the squid's situation, where it reaches a speed of 2.50 m/s, its kinetic energy acts as a sudden burst of energy to enhance its chances of escape. Efficient conversion of stored potential energy into kinetic energy helps it propel forward rapidly, which is crucial for survival against predators.

Animal Propulsion

Animal propulsion is how creatures like squids and octopuses move or propel themselves. Unlike humans, they utilize a unique method of expelling water from their bodies to generate movement. This action results in a rapid burst of speed, allowing them to swiftly evade threats or catch prey.

For squids and octopuses, they store a certain amount of water inside their bodies. When threatened, they contract muscle cavities to force the water out through a narrow opening. This action propels them forward in a swift motion.

• This is quite similar to how rockets work, as they expel gas to create thrust.
• This type of propulsion is a stunning example of how simple mechanical strategies can be employed by marine life.

The squid, for instance, achieves significant motion by combining water ejection and body streamlining. Their ability to modulate water expulsion speed is vital for effective animal propulsion.

Momentum Equation

The momentum equation is a critical principle in physics that helps us understand how motion is transferred in a system of objects. Momentum itself is a measure of the quantity of motion an object

The conservation of momentum states that in a closed system, the total momentum before an event is equal to the total momentum after the event, provided no external forces are acting on it.

• Initial Momentum = Final Momentum

In mathematical terms, it is often expressed as:

• \(m_1 v_1 + m_2 v_2 = m_1 u_1 + m_2 u_2\)

For our squid, this means that the momentum of the expelled water is equal but opposite to the momentum of the squid, resulting in movement. This exquisite balance allows the squid to reach a desired speed effectively, helping it in defensive maneuvers against predators.