A photon of red light (wavelength = 680 nm) and a Ping-Pong ball (mass = 1.90 × 10-3 kg) have the same momentum. At what speed is the ball moving?
The correct answer and explanation is:
To find the speed of the Ping-Pong ball, we use the fact that both the photon and the ball have the same momentum. The momentum of a photon is given by: p=hλp = \frac{h}{\lambda}
Where:
- pp is momentum
- h=6.626×10−34 J\cdotpsh = 6.626 \times 10^{-34} \, \text{J·s} (Planck’s constant)
- λ=680 nm=680×10−9 m\lambda = 680 \, \text{nm} = 680 \times 10^{-9} \, \text{m}
Step 1: Calculate the momentum of the photon
p=6.626×10−34680×10−9=9.74×10−28 kg\cdotpm/sp = \frac{6.626 \times 10^{-34}}{680 \times 10^{-9}} = 9.74 \times 10^{-28} \, \text{kg·m/s}
Step 2: Use this momentum to find the speed of the Ping-Pong ball
Momentum is also defined for classical objects as: p=mvp = mv
Solving for velocity vv: v=pm=9.74×10−281.90×10−3=5.13×10−25 m/sv = \frac{p}{m} = \frac{9.74 \times 10^{-28}}{1.90 \times 10^{-3}} = 5.13 \times 10^{-25} \, \text{m/s}
Final Answer:
5.13×10−25 m/s\boxed{5.13 \times 10^{-25} \, \text{m/s}}
Explanation (300 words):
To compare the motion of a photon and a Ping-Pong ball, we must use the concept of momentum. A photon, which has no mass, still carries momentum. Its momentum is calculated using Planck’s constant and its wavelength. In this case, the red light photon has a wavelength of 680 nanometers, which we convert to meters to keep all units consistent in the SI system.
By dividing Planck’s constant by the wavelength, we find the photon’s momentum to be approximately 9.74×10−28 kg\cdotpm/s9.74 \times 10^{-28} \, \text{kg·m/s}. Since the problem states that the Ping-Pong ball has the same momentum as the photon, we set the two momenta equal.
To find the ball’s speed, we apply the classical formula for momentum, which is mass times velocity. Rearranging the formula to solve for velocity, we divide the momentum by the ball’s mass. The Ping-Pong ball has a mass of 1.90×10−3 kg1.90 \times 10^{-3} \, \text{kg}, so its velocity becomes incredibly small, about 5.13×10−25 m/s5.13 \times 10^{-25} \, \text{m/s}.
This extremely low speed shows how large and massive objects must move very slowly to match the tiny momentum of a single photon. The result also emphasizes how different quantum objects like photons are from macroscopic objects like a ball. Despite having the same momentum, the scale of motion between them differs greatly due to the massive difference in mass.