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The image contains two physics problems related to partial derivatives and total differentials. Let’s analyze and solve the questions:

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### Problem 1:
Given that momentum \( p = mv \), where:
- \( m \) is the mass of an object,
- \( v \) is the velocity.

The force acting on an object is related to the rate of change of momentum:
\[ F = \frac{dp}{dt}. \]

1a) If the mass \( m \) is constant, show that \( F = ma \), where \( a \) is the acceleration (\( \frac{dv}{dt} \)).

**Solution:**
If the mass is constant, we can differentiate \( p = mv \) with respect to time:
\[
F = \frac{dp}{dt} = \frac{d}{dt}(mv) = m \frac{dv}{dt} = ma,
\]
where \( \frac{dv}{dt} \) is the acceleration \( a \). Thus, \( F = ma \) is proven.

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1b) If the mass varies with time, given that \( m = m_0 e^{-bt} \), find the force \( F \).

**Solution:**
Here, we have \( p = m(t) v \), where \( m(t) = m_0 e^{-bt} \). The force is given by the time derivative of the momentum:
\[
F = \frac{dp}{dt} = \frac{d}{dt}(m(t)v) = \frac{dm}{dt} v + m(t) \frac{dv}{dt}.
\]
Now, find \( \frac{dm}{dt} \):
\[
\frac{dm}{dt} = \frac{d}{dt}(m_0 e^{-bt}) = -b m_0 e^{-bt} = -b m.
\]
Thus, the force is:
\[
F = (-b m)v + m a = -b mv + ma.
\]

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### Problem 2:
The coefficient of thermal expansion for a gas is defined as:
\[
\beta = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_P,
\]
where \( V \) is the volume, and \( T \) is the temperature. 

2a) Determine \( \beta \) for a gas described by the equation \( P(V - b) = RT \).

**Solution:**
The equation is a modified version of the ideal gas law:
\[
P(V - b) = RT,
\]
where \( b \) is a constant. To find \( \beta \), we first solve for \( V \) as a function of \( T \):
\[
V = \frac{RT}{P} + b.
\]
Now, differentiate \( V \) with respect to \( T \) while keeping \( P \) constant:
\[
\frac{\partial V}{\partial T} = \frac{R}{P}.
\]
Substitute into the definition of \( \beta \):
\[
\beta = \frac{1}{V} \left( \frac{R}{P} \right).
\]
Thus, \( \beta = \frac{R}{P(V)} \).

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2b) Analyze the second equation \( \left( p + \frac{a}{v^2} \right)(v - b) = RT \), where \( v = \frac{V}{n} \), and constants are \( a \), \( b \), and \( R \).

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Would you like further details on any step? Here are five related questions to deepen your understanding:

1. How does varying mass influence the net force on an object in non-relativistic mechanics?
2. What assumptions are made about gas particles when applying the ideal gas law or van der Waals equation?
3. How do you interpret the constant \( b \) in the context of real gas behavior?
4. In which physical scenarios would the expression \( F = -b mv + ma \) be most applicable?
5. How does the thermal expansion coefficient change with pressure for an ideal gas?

**Tip:** The Van der Waals equation accounts for intermolecular forces and finite molecular sizes, refining the ideal gas law to more accurately describe real gases.

1. a) If the mass of an object does not change with time, prove that F = ma, where a is the acceleration (dv/dt). b) If the mass of an object changes with time according to m = m_0e^{-bt}, find F. 2. Find the coefficient of thermal expansion β for gases with given equations: a) P(V - b) = RT; b) (p + a/v^2)(v - b) = RT.

Explore the application of partial derivatives and total differentials in physics. This problem set covers momentum, force in varying mass systems, and the coefficient of thermal expansion for gases. Ideal for undergraduate physics students.

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