Problem Analysis

We are tasked to determine the correct relationship between force $F$ and time $t$, given the following:

1. A force $F$ acts on a body of mass $m = 30 \, \text{kg}$ over $8 \, \text{seconds}$.
2. The change in velocity ($\Delta v$) within $8 \, \text{seconds}$ equals $72 \, \text{km/h}$, which we need to convert to $\text{m/s}$:
   $\Delta v = 72 \, \text{km/h} = \frac{72 \times 1000}{3600} = 20 \, \text{m/s}.$
3. Using the relationship between force, mass, and acceleration ($F = ma$), we calculate the acceleration from the change in velocity and use it to find the correct force graph.

Step-by-Step Solution

Step 1: Calculate Acceleration ($a$)

The acceleration is determined by: $a = \frac{\Delta v}{\Delta t} = \frac{20}{8} = 2.5 \, \text{m/s}^2.$

Step 2: Relating Force and Acceleration

Using $F = ma$, the magnitude of the force is: $F = 30 \times 2.5 = 75 \, \text{N}.$ This is the average force over the 8 seconds. The variation in force over time will depend on how the velocity changes (linearly or otherwise).

Step 3: Analyze the Given Options

We analyze each graph based on the information:

• (a) Shows a force increasing linearly for 2 seconds, remaining constant for 4 seconds, and then decreasing linearly for 2 seconds.
  This implies acceleration changes non-linearly, which is inconsistent with the problem.

• (b) Shows a force increasing linearly with time.
  If $F \propto t$, acceleration would increase continuously, which is inconsistent with constant $\Delta v$ over $8 \, \text{s}$.

• (c) Shows a force decreasing linearly over time.
  This is inconsistent with constant $\Delta v$ and average force $75 \, \text{N}$.

• (d) Shows a constant force of 75 N over 8 seconds, which aligns with constant acceleration of $2.5 \, \text{m/s}^2$ and $\Delta v = 20 \, \text{m/s}$.
  Thus, (d) is the correct choice.

Final Answer

The correct graph is (d).

Would you like a detailed explanation of the concepts or any follow-up?

Relative Questions

1. How does the relationship $F = ma$ explain motion with constant acceleration?
2. Why is $\Delta v$ converted to $\text{m/s}$ from $\text{km/h}$?
3. What does a linearly increasing force indicate about acceleration?
4. How is the area under the force-time graph related to momentum?
5. How do we interpret average force in problems involving changing forces?

Tip

When working with graphs, always connect them to physical quantities like velocity, acceleration, and their rates of change over time.