A honeybee with a mass of 0.175 g lands on one end of a popsicle stick. After sitting at rest for a moment, the bee runs toward the other end with a velocity 1.41 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 need to understand the concept of conservation of momentum. The total momentum before the bee runs on the stick should be equal to the total momentum after the bee starts moving.
Momentum (p) is calculated by multiplying mass (m) with velocity (v), and it is a vector quantity, meaning it has both magnitude and direction. In this case, the magnitude of the bee's momentum before and after running on the stick will be the same because there are no external forces acting on the system.
Let's calculate the initial momentum (p1) of the bee before it starts running. The mass of the bee is 0.175 g, which converts to 0.175/1000 = 0.000175 kg.
The initial velocity (v1) of the bee is taken as 0 because it is at rest.
So, the initial momentum (p1) of the bee is given by:
p1 = m1 * v1
= 0.000175 kg * 0 m/s
= 0 kg·m/s
Now, let's calculate the final momentum (p2) of the bee and the stick together after the bee starts running. The mass of the stick is 5.20 g, which converts to 5.20/1000 = 0.0052 kg.
The final velocity (v2) of the bee and the stick together is given as 1.41 cm/s in the negative direction. We need to convert this to meters per second (m/s).
1 cm = 0.01 m (converting cm to meters)
So, 1.41 cm/s = 1.41 * 0.01 m/s = 0.0141 m/s (final velocity in m/s)
Now, calculating the final momentum (p2) of the bee and the stick together:
p2 = (m1 + m2) * v2
= (0.000175 kg + 0.0052 kg) * (-0.0141 m/s)
= 0.005375 kg * (-0.0141 m/s)
= -0.000075 kg·m/s
Since momentum is conserved (p1 = p2), the initial momentum of the bee should be equal to the final momentum of the bee and the stick.
Therefore, we have:
0 kg·m/s = -0.000075 kg·m/s
The negative sign on the final momentum indicates that the direction is opposite to the initial direction of the bee. Thus, the speed of the 5.20 g stick relative to the water is 0.0141 m/s.