A honeybee with a mass of 0.175 g lands on one end of a popsicle stick (Active Example 9-2). After sitting at rest for a moment, the bee runs toward the other end with a velocity 1.47 cm/s relative to the still water. What is the speed of the 5.20 g stick relative to the water? (Assume the bee's motion is in the negative direction.
To solve this problem, we can use the principle of conservation of momentum. The initial momentum of the system, consisting of the bee and the stick, is equal to the final momentum of the system.
Step 1: Write down the given information.
- Mass of the bee (m1) = 0.175 g = 0.175 g/1000 g (since 1 g = 0.001 kg)
- Velocity of the bee (v1) = -1.47 cm/s (since the bee's motion is in the negative direction)
- Mass of the stick (m2) = 5.20 g = 5.20 g/1000 g (since 1 g = 0.001 kg)
- Velocity of the stick (v2) = ? (what we need to find)
Step 2: Convert the masses to kilograms.
- m1 = (0.175 g/1000 g) = 0.000175 kg
- m2 = (5.20 g/1000 g) = 0.00520 kg
Step 3: Apply the conservation of momentum principle.
The initial momentum (p_initial) is equal to the final momentum (p_final).
- Initial momentum = m1 * v1
- Final momentum = (m1 + m2) * v2
So we can write the equation as:
m1 * v1 = (m1 + m2) * v2
Step 4: Rearrange the equation and solve for v2.
v2 = (m1 * v1) / (m1 + m2)
Substituting the given values:
v2 = (0.000175 kg * -1.47 cm/s) / (0.000175 kg + 0.00520 kg)
Step 5: Calculate the value of v2.
v2 = -0.00025725 kg*cm/s / 0.005375 kg
v2 ≈ -0.0479 cm/s
The speed of the 5.20 g stick relative to the water is approximately 0.0479 cm/s in the negative direction.
To find the speed of the stick relative to the water, we need to take into account the motion of both the bee and the stick. Let's break down the problem step by step:
1. First, we need to determine the initial velocity of the stick before the bee starts moving. The problem states that the bee has a velocity of 1.47 cm/s relative to the still water. Since the bee's motion is in the negative direction, the stick's initial velocity will be -1.47 cm/s.
2. Now, we need to consider the mass and initial velocity of the stick. The stick has a mass of 5.20 g. However, the problem does not provide the stick's initial velocity relative to the water. We need this information to calculate the stick's speed relative to the water.
3. To find the stick's initial velocity, we can make use of the conservation of momentum. Since there are no external forces acting on the system (bee + stick), the total momentum before and after the bee starts running should be the same.
Before the bee starts running:
Total initial momentum = bee's momentum + stick's momentum
Since the bee is initially at rest, its momentum is zero. The momentum of an object is given by the product of its mass and velocity.
Total initial momentum = (bee's mass x bee's velocity) + (stick's mass x stick's initial velocity)
Total initial momentum = 0 + (5.20 g x stick's initial velocity)
4. The total momentum after the bee starts running is the product of the combined mass of the bee and the stick and their resultant velocity.
Total final momentum = (bee + stick) mass x resultant velocity
Since the bee is running toward the other end of the stick, the resultant velocity of the system will be negative because the bee's motion is in the negative direction.
Total final momentum = (bee + stick) mass x (-resultant velocity)
Equating the total initial momentum to the total final momentum, we have:
0 + (5.20 g x stick's initial velocity) = (bee + stick) mass x (-resultant velocity)
5. Now, we can solve for the stick's initial velocity by substituting the values into the equation:
5.20 g x stick's initial velocity = (0.175 g + 5.20 g) x (-1.47 cm/s)
Simplifying the equation:
5.20 g x stick's initial velocity = 5.375 g x (-1.47 cm/s)
6. To find the stick's initial velocity, we divide both sides of the equation by 5.20 g:
stick's initial velocity = (5.375 g x (-1.47 cm/s)) / 5.20 g
7. The resulting velocity will have units of cm/s, but we are interested in the speed, which is the magnitude of velocity. Since velocity is a vector quantity, we take the absolute value of the velocity to obtain the speed:
speed of the stick relative to the water = |stick's initial velocity|
By following these steps and calculating the final equation, you should be able to find the speed of the stick relative to the water.