Problem 78 - College Physics

Question

Archerfish are tropical fish that hunt by shooting drops of water from their mouths at insects above the water's surface to knock them into the water, where the fish can eat them. \(\mathrm{A} 65 \mathrm{~g}\) fish at rest just on the water's surface can expel a \(0.30 \mathrm{~g}\) drop of water in a short burst of \(5.0 \mathrm{~ms}\). High-speed measurements show that the water has a speed of \(2.5 \mathrm{~m} / \mathrm{s}\) just after the archerfish expels it. A fish shoots a drop of water at an insect that hovers on the water's surface. Just before colliding with the insect, the drop is still moving at the speed it had when it left the fish's mouth. In the collision, the drop sticks to the insect, and the speed of the insect and water just after the collision is measured to be \(2.0 \mathrm{~m} / \mathrm{s}\). What is the insect's mass? A. \(0.038 \mathrm{~g}\) B. \(0.075 \mathrm{~g}\) C. \(0.24 \mathrm{~g}\) D. \(0.38 \mathrm{~g}\)

Step-by-Step Solution

Answer

The insect's mass is B. 0.075 g.

Step 1: Understanding the Conservation of Momentum

When two objects collide, their total momentum just before the collision is equal to their total momentum just after the collision, assuming no external forces. We will use this principle to calculate the mass of the insect. The formula for momentum is \( p = mv \), where \( m \) is mass and \( v \) is velocity.

Step 2: Establishing the Initial Momentum

Initially, the insect is not moving, so it has no momentum. The initial momentum of the system comes entirely from the drop of water. Calculate this using the mass of the water drop and its velocity. \[ p_{\text{initial}} = m_{\text{water}} \times v_{\text{water}} \]\[ p_{\text{initial}} = 0.30 \times 10^{-3} \text{ kg} \times 2.5 \text{ m/s} \]

Step 3: Calculating the Initial Momentum

Perform the calculation using the masses and velocities:\[ p_{\text{initial}} = 0.30 \times 10^{-3} \times 2.5 = 0.75 \times 10^{-3} \text{ kg m/s} \]

Step 4: Formulating the Final Momentum

The final momentum includes both the insect and the water moving together with a new velocity. The formula is:\[ p_{\text{final}} = (m_{\text{water}} + m_{\text{insect}}) \times v_{\text{final}} \] \[ p_{\text{final}} = (0.30 \times 10^{-3} + m_{\text{insect}}) \times 2.0 \]

Step 5: Setting initial and final momentum equal

Since momentum is conserved, set the initial and final momentum equal to each other:\[ 0.75 \times 10^{-3} = (0.30 \times 10^{-3} + m_{\text{insect}}) \times 2.0 \]

Step 6: Solving for Insect Mass

Isolate \( m_{\text{insect}} \) in the equation:\[ 0.75 \times 10^{-3} = 0.60 \times 10^{-3} + 2.0 \times m_{\text{insect}} \]Subtract \( 0.60 \times 10^{-3} \) from both sides:\[ 0.15 \times 10^{-3} = 2.0 \times m_{\text{insect}} \]Divide by \( 2.0 \):\[ m_{\text{insect}} = 0.075 \times 10^{-3} \text{ kg} = 0.075 \text{ g} \]

Step 7: Verification

Check if the calculated mass fits one of the given options and makes sense, verifying that there are no calculation errors or missed considerations in initial assumptions.

Key Concepts

Physics Problem Solving

Physics problem-solving is all about breaking a physics problem into manageable steps, using given data to find unknowns. It involves understanding the underlying principles that govern the problem, like the conservation of momentum. Knowing how to set up equations correctly is crucial in this process. By analyzing what is given and what needs to be found, students can create a structured path to the solution. This methodical approach helps in avoiding errors and ensures each aspect of the problem is considered.
In the case of the archerfish and the insect, the problem solving begins with identifying the objects at play — the water drop and the insect — and establishing the interaction, which is a collision. By leveraging the conservation of momentum principle, the initial and final states of this interaction can be compared mathematically to solve for the unknown, which is the mass of the insect.

Momentum Calculation

Momentum is calculated as the product of an object's mass and velocity, expressed as \( p = mv \). This fundamental concept in physics allows us to understand how objects in motion interact during collisions.
To find the momentum of the water drop in the archerfish problem, we use its mass and velocity before the collision. Here, the water drop's mass is converted from grams to kilograms, since standard units are crucial in physics calculations:

• Mass of water: \( 0.30 \text{ g} = 0.30 \times 10^{-3} \text{ kg} \)
• Velocity of water: \( 2.5 \text{ m/s} \)
• Initial Momentum: \( p_{\text{initial}} = 0.30 \times 10^{-3} \times 2.5 \text{ m/s} \)

This approach ensures each step of the calculation is clear and accurate, providing a foundation for finding the solution to a problem involving changes in momentum.

Collision and Impact

In the context of collisions, conservation of momentum is key. A collision involves two or more bodies exerting forces upon each other for a short period. In our exercise, the water drop collides with the insect, with the two sticking together afterwards, known as a perfectly inelastic collision.
Such collisions are characterized by a conservation of total momentum, even when kinetic energy isn’t conserved. The combined system of water and insect will share a common velocity post-impact. This results in the momentum equation \( (m_{\text{water}} + m_{\text{insect}}) \times v_{\text{final}} = p_{\text{initial}} \). Solving this equation gives insight into how momentum was distributed between the two masses.

Physics of Motion

The physics of motion explores how objects move and interact under various forces. In the archerfish scenario, understanding motion includes analyzing both translational speeds and the effects of force during the impact.
The initial speed of the expelled water drop forms part of the projectile motion display by the archerfish. Since the water was manipulated through a high-speed spurt, it imparted enough kinetic energy to cause the insect to move upon collision.
This motion analysis is crucial in determining the insect's mass: the system's earlier motion translated into an impact, affecting the insect's speed post-collision, combined with the precise velocity calculations, facilitates understanding how movement in different bodies correlates within a physical system. Understanding such interactions is foundational in many physics applications, from engineering to natural phenomena.