A 96 kg boat that is 6.6 m in length is initially 7.9 m from the pier. A 35 kg child stands at the end of the boat closest to the pier. The child then notices a turtle on a rock at the far end of the boat and proceeds to walk to the far end of the boat to observe the turtle. 7.9 m 6.6 m How far is the child from the pier when she reaches the far end of the boat? Assume there is no friction between boat and water. Answer in units of m.

To solve this problem, we can use the principle of conservation of momentum. The system (boat + child) does not experience any external horizontal forces, so the center of mass (COM) of the system must remain stationary.

### Given:
- Mass of the boat: \( M_b = 96 \, \text{kg} \)
- Mass of the child: \( M_c = 35 \, \text{kg} \)
- Length of the boat: \( L = 6.6 \, \text{m} \)
- Initial distance from the pier to the boat: \( 7.9 \, \text{m} \)

### Step 1: Position of the center of mass (COM) of the system initially.

We assume the boat is initially stationary. The position of the COM of the system can be calculated using the formula for the center of mass:

\[
X_{\text{COM}} = \frac{M_b x_b + M_c x_c}{M_b + M_c}
\]

- Let \( x_b \) be the position of the center of mass of the boat (which is at its geometric center, halfway along its length). Since the boat is 6.6 m long and one end is 7.9 m away from the pier, the center of the boat is:

\[
x_b = 7.9 + \frac{6.6}{2} = 7.9 + 3.3 = 11.2 \, \text{m}
\]

- Let \( x_c \) be the initial position of the child. The child starts at the end of the boat closest to the pier, which is 7.9 m from the pier. Thus, \( x_c = 7.9 \, \text{m} \).

Now we calculate the initial COM:

\[
X_{\text{COM, initial}} = \frac{96(11.2) + 35(7.9)}{96 + 35} = \frac{1075.2 + 276.5}{131} = \frac{1351.7}{131} \approx 10.32 \, \text{m}
\]

### Step 2: Final position of the child and boat.

When the child walks to the far end of the boat, the boat moves in the opposite direction due to the lack of external forces. Let \( \Delta x_b \) be the displacement of the boat (relative to the water) and \( \Delta x_c \) be the displacement of the child (relative to the boat). The child moves the full length of the boat, so:

\[
\Delta x_c = 6.6 \, \text{m}
\]

Since the center of mass remains stationary, we know:

\[
M_b \Delta x_b + M_c (-\Delta x_c) = 0
\]

This gives:

\[
96 \Delta x_b = 35(6.6)
\]

Solving for \( \Delta x_b \):

\[
\Delta x_b = \frac{35(6.6)}{96} = \frac{231}{96} \approx 2.406 \, \text{m}
\]

Thus, the boat moves 2.406 m away from the pier as the child walks.

### Step 3: Final position of the child relative to the pier.

The child’s final position is the sum of her initial position relative to the boat, the distance she walks on the boat, and the movement of the boat. Since the child starts at 7.9 m and walks the length of the boat, but the boat itself moves, we need to account for both movements:

\[
x_{\text{child, final}} = 7.9 + 6.6 - 2.406 = 14.5 - 2.406 = 12.094 \, \text{m}
\]

Thus, the child is approximately **12.09 m** from the pier when she reaches the far end of the boat.

---

Do you want further clarification on any step? Or additional questions?

Here are 5 related questions you might explore:
1. What happens to the system if there is friction between the boat and water?
2. How would the result change if the child walked at a different speed?
3. What would the final position be if the boat had a different mass?
4. How does the principle of conservation of momentum apply to other similar systems?
5. How would the motion differ if there were another person on the boat?

**Tip:** In problems involving conservation of momentum, always account for the center of mass and external forces acting on the system.

A 96 kg boat that is 6.6 m in length is initially 7.9 m from the pier. A 35 kg child stands at the end of the boat closest to the pier. The child then notices a turtle on a rock at the far end of the boat and proceeds to walk to the far end of the boat to observe the turtle. How far is the child from the pier when she reaches the far end of the boat? Assume there is no friction between boat and water. Answer in units of m.

Solve the problem of a 35 kg child walking across a 96 kg boat using the conservation of momentum principle. Learn how to calculate the final position of the child relative to the pier by applying the center of mass formula. Suitable for high school physics students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation