A 63 kg canoeist stands in the middle of her 19 kg canoe. The canoe is 3.0 m long, and the end that is closest to land is 2.5 m from the shore. The canoeist now walks toward the shore until she comes to the end of the canoe.

a. When the canoeist stops at the end of her canoe, is her distance from the shore equal to, greater than, or less than 2.5 m?

b. Verify your answer to part (a) by calculating the distance from the canoeist to shore.

The center of gravity of the system will not move (ignoring water resistance).

It remains 2.5 + 3/2 = 4 m from shore
The canoe, being lighter, moves back more than the canoeist moves forward.
Call final distance of person from shore x
total mass = 63+19 = 82 kg
82 kg * 4 m = 328 kg m
so
328 = 63 x + 19 (x+1.5)
328 = 63 x + 19 x + 28.5
299.5 = 82 x
x = 3.65 m
This is not news to those of use who fool around in small boats.

a. When the canoeist stops at the end of her canoe, her distance from the shore is less than 2.5 m.

b. To calculate the distance from the canoeist to shore, we can use the principle of conservation of momentum.

Before the canoeist starts moving, the total momentum of the canoe and the canoeist is zero, since there is no net external force acting on them. We can express this as:

(m1v1) + (m2v2) = 0

where:
m1 = mass of the canoeist = 63 kg
v1 = velocity of the canoeist
m2 = mass of the canoe = 19 kg
v2 = velocity of the canoe

When the canoeist starts moving towards the shore, the canoe and the canoeist will move in the opposite direction. We can consider the moment before and after the canoeist starts moving.

Before the canoeist starts moving, the canoe and the canoeist's velocity are both 0. After the canoeist starts moving, the canoe and the canoeist have opposite velocities, so we can write:

(m1v1) + (m2v2) = 0

Solving for v2, we get:

v2 = - (m1v1) / m2

Now we can calculate the distance from the canoeist to the shore using the formula for velocity:

v = ∆x / ∆t

where:
v = velocity
∆x = change in position
∆t = change in time

Since we are interested in the distance from the canoeist to the shore, we can consider the moment the canoeist stops.

When the canoeist stops, the velocities of the canoe and the canoeist are equal in magnitude and opposite in direction. We can write:

v1 = -v2

Using this relationship, we can rewrite the equation for v2 as:

v2 = - (m1v1) / m2 = - (m1(-v1)) / m2 = (m1v1) / m2

Now we can calculate the distance from the canoeist to the shore using the formula for velocity:

v = ∆x / ∆t

Since the canoeist has stopped, the velocity of the canoeist is 0, so we have:

0 = ∆x / ∆t

This means that the distance from the canoeist to the shore (∆x) is 0.

Therefore, the distance from the canoeist to the shore when she stops at the end of the canoe is 0.

To answer part (a) of the question, we need to consider the concept of center of mass. When the canoeist is standing in the middle of the canoe, the center of mass of the canoe-canoeist system is also in the middle. This means that the weight of the system is evenly distributed, and the position of the center of mass is also in the middle of the canoe.

When the canoeist walks towards the end of the canoe, the center of mass of the system will shift towards that end. Since the canoeist is much heavier compared to the canoe, the center of mass will be closer to the canoeist's end.

Therefore, the distance between the canoeist and the shore when she stops at the end of the canoe will be closer to the end of the canoe and therefore less than 2.5 m.

To verify this, we can calculate the distance from the canoeist to the shore in part (b) of the question.

To calculate the distance from the canoeist to the shore, we can use the concept of the center of mass. The center of mass in this case is given by:

**CM = (m_canoe * x_canoe + m_canoeist * x_canoeist) / (m_canoe + m_canoeist)**

Where:
m_canoe is the mass of the canoe (19 kg),
m_canoeist is the mass of the canoeist (63 kg),
x_canoe is the initial distance between the shore and the end of the canoe (2.5 m),
x_canoeist is the distance the canoeist moved towards the shore.

Substituting the given values into the formula:

**CM = (19 kg * 2.5 m + 63 kg * 0 m) / (19 kg + 63 kg)**
**CM = (47.5 kg + 0 kg) / 82 kg**
**CM = 0.579 m**

Therefore, the center of mass is at a distance of 0.579 m from the shore. Since the canoeist stops at the end of the canoe, the distance from the canoeist to the shore is equal to the distance from the center of mass to the shore. Therefore, the distance from the canoeist to the shore is also 0.579 m.

This calculation verifies that the distance from the canoeist to the shore is less than 2.5 m, as determined in part (a) of the question.

HOU VAN JULLIE