The problem in the image involves a block sliding along a flat horizontal surface with an initial velocity of $v_0 = 2.0 \, \text{m/s}$ and experiencing a constant resistive force of $F = 4.0 \, \text{N}$, which brings the block to rest after covering a distance $d = 2.0 \, \text{m}$.

We are asked to find the mass $m$ of the block.

Step-by-step solution:

1. Using the work-energy principle:
   The work done by the resistive force brings the block to rest. According to the work-energy theorem: $W = \Delta K$ where $W$ is the work done by the force, and $\Delta K$ is the change in kinetic energy.

   The initial kinetic energy $K_i$ is: $K_i = \frac{1}{2} m v_0^2$ Since the block comes to rest, the final kinetic energy $K_f = 0$. Therefore, the change in kinetic energy is: $\Delta K = K_f - K_i = 0 - \frac{1}{2} m v_0^2 = -\frac{1}{2} m v_0^2$

2. Calculating the work done by the resistive force:
   The work done by a constant force is given by: $W = F \cdot d$ where $F = 4.0 \, \text{N}$ and $d = 2.0 \, \text{m}$. Therefore: $W = 4.0 \, \text{N} \times 2.0 \, \text{m} = 8.0 \, \text{J}$

3. Equating the work done to the change in kinetic energy:
   According to the work-energy theorem, the work done by the force is equal to the negative of the change in kinetic energy: $-\frac{1}{2} m v_0^2 = W$ Substituting the values: $-\frac{1}{2} m (2.0 \, \text{m/s})^2 = -8.0 \, \text{J}$ Simplifying: $\frac{1}{2} m \times 4 = 8.0$ $2m = 8.0$ $m = 4.0 \, \text{kg}$

Final Answer:

The mass of the block is $\boxed{4.0 \, \text{kg}}$.

Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. What is the significance of the work-energy theorem in mechanics?
2. How does the resistive force impact the motion of objects?
3. Can you derive the formula for work done by a force using basic principles of energy?
4. What would happen to the stopping distance if the resistive force doubled?
5. How does this problem change if the surface is inclined?

Tip: Always ensure that all forces acting on a body are accounted for when solving problems involving work and energy.