To solve this problem, we need to find the magnitude of the average force the rock experiences while it is in the water. Here’s the step-by-step process:

Step 1: Determine the velocity of the rock just before it hits the water

The rock is dropped from a height $h = 3.9 \, \text{m}$, and we can find its velocity just before impact using the equation of motion under free fall:

$v = \sqrt{2gh}$

Where:

• $g = 9.81 \, \text{m/s}^2$ (acceleration due to gravity)
• $h = 3.9 \, \text{m}$

$v = \sqrt{2 \times 9.81 \, \text{m/s}^2 \times 3.9 \, \text{m}} = \sqrt{76.518} \approx 8.75 \, \text{m/s}$

So, the velocity of the rock just before it hits the water is approximately $8.75 \, \text{m/s}$.

Step 2: Find the velocity of the rock after 1.96 seconds in water

The problem states that the velocity decreases by 50% after $t = 1.96 \, \text{s}$. So, the velocity after 1.96 seconds is:

$v_{\text{after}} = 0.5 \times 8.75 \, \text{m/s} = 4.375 \, \text{m/s}$

Step 3: Calculate the deceleration (negative acceleration)

The deceleration (or negative acceleration) can be found using the formula:

$a = \frac{\Delta v}{\Delta t}$

Where:

• $\Delta v = v_{\text{after}} - v_{\text{initial}} = 4.375 \, \text{m/s} - 8.75 \, \text{m/s} = -4.375 \, \text{m/s}$
• $\Delta t = 1.96 \, \text{s}$

$a = \frac{-4.375 \, \text{m/s}}{1.96 \, \text{s}} \approx -2.23 \, \text{m/s}^2$

Step 4: Apply Newton's second law to find the average force

Newton's second law states that the force is the product of mass and acceleration:

$F = ma$

Where:

• $m = 13 \, \text{kg}$
• $a \approx -2.23 \, \text{m/s}^2$

$F = 13 \, \text{kg} \times -2.23 \, \text{m/s}^2 \approx -28.99 \, \text{N}$

Step 5: Magnitude of the average force

Since the problem asks for the magnitude, we take the absolute value:

$F_{\text{avg}} = | -28.99 \, \text{N} | \approx 29 \, \text{N}$

Final Answer:

The magnitude of the average force the rock experiences during the time $t$ is approximately 29 N.

Would you like more details or have any questions about the solution?

Here are some related questions:

1. How would the force change if the mass of the rock was doubled?
2. What would the force be if the velocity reduction was only 25%?
3. How would the result change if the time interval was 3 seconds instead of 1.96 seconds?
4. What is the significance of deceleration in this context?
5. How would air resistance affect the velocity before hitting the water?

Tip: Always remember that when an object falls freely, its acceleration is due to gravity and independent of its mass.