A great white shark of mass m1, coasting through the water in a nearly frictionless way at speed v1 engulfs a tuna of mass m2=0.10 m1 traveling in the same direction at speed v2 swallowing it in one bite.

a) What is the speed of the shark after its meal?

b) Did the shark gain (kinetic) energy, lose energy, or have its energy remain the same in the process?

(a)

Law of conservation of momentum:
m1v1+m2v2=(m1+m2)v
since m2=0.1m1
m1v1+0.1m1v2=(1.1m1)v
solve for v
v=(m1)(v1+0.1v2)/(1.1m1)
=(v1+0.1v2)/1.1

(b)
Assuming no loss in total energy,
and assuming v1>v2, then
the shark itself (without food) has lost energy, since its velocity has reduced.
If the food becomes part of the shark, then the shark has gained kinetic energy, since total energy has not changed.

To solve this problem, we can use the principle of conservation of momentum. According to this principle, the total momentum before the interaction between the shark and the tuna is equal to the total momentum after the interaction.

Let's denote the speed of the shark after the meal as v'1. We can use the equation for conservation of momentum:

(m1 * v1) + (m2 * v2) = (m1 * v'1)

a) Speed of the shark after the meal (v'1):

Substituting the known values:

(m1 * v1) + (0.10 * m1 * v2) = (m1 * v'1)

Since the shark engulfs the tuna in one bite, their masses combine. Thus, the equation becomes:

(m1 * v1) + (0.10 * m1 * v2) = (m1 * v'1)

Simplifying, we have:

v1 + (0.10 * v2) = v'1

Therefore, the speed of the shark after the meal is v'1 = v1 + (0.10 * v2).

b) Energy change of the shark:

To determine if the shark gained, lost, or remained the same in terms of kinetic energy, we need to analyze the kinetic energy before and after. The equation for kinetic energy is:

Kinetic energy = (1/2) * Mass * Velocity^2

Since we know that both the mass of the shark and the tuna are now combined, the initial kinetic energy before the meal is:

Initial Kinetic energy = (1/2) * (m1 + m2) * v1^2

After the meal, the kinetic energy of the shark is:

Final Kinetic energy = (1/2) * (m1) * v'1^2

Comparing these two expressions, we can see that:

Final Kinetic energy - Initial Kinetic energy = (1/2) * (m1) * v'1^2 - (1/2) * (m1 + m2) * v1^2

Simplifying further:

Final Kinetic energy - Initial Kinetic energy = (1/2) * m1 * [(v1 + (0.10 * v2))^2 - v1^2]

Expanding and simplifying:

Final Kinetic energy - Initial Kinetic energy = (1/2) * m1 * [v1^2 + 2 * 0.10 * v1 * v2 + (0.10 * v2)^2 - v1^2]

Final Kinetic energy - Initial Kinetic energy = (1/2) * m1 * [0.01 * v2^2 + 0.20 * v1 * v2]

The result depends on the values of v1 and v2. If v1 and v2 are both positive, then Final Kinetic energy - Initial Kinetic energy will also be positive, indicating that the shark gained kinetic energy. If v1 and v2 have opposing signs (one positive and the other negative), then Final Kinetic energy - Initial Kinetic energy can be negative, indicating that the shark lost kinetic energy. If v1 and v2 are both negative, then Final Kinetic energy - Initial Kinetic energy may be positive or negative, depending on their magnitudes.

So, the energy change of the shark depends on the specific values of v1 and v2.