A photon of red light (wavelength = 690 nm) and a Ping-Pong ball (mass = 2.60 A1 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 will first calculate the momentum of the photon and equate it to the momentum of the Ping-Pong ball. The formula for momentum ( p ) is:
[
p = mv
]
where:
- ( m ) is the mass of the object (for the ball),
- ( v ) is its velocity (which we want to solve for).
The momentum of a photon is given by the relation:
[
p = \frac{h}{\lambda}
]
where:
- ( h ) is Planck’s constant (( 6.626 \times 10^{-34} \, \text{J·s} )),
- ( \lambda ) is the wavelength of the light.
Step 1: Calculate the momentum of the photon
Given:
- Wavelength ( \lambda = 690 \, \text{nm} = 690 \times 10^{-9} \, \text{m} )
We use the formula for the photon’s momentum:
[
p_{\text{photon}} = \frac{h}{\lambda}
]
Substitute the known values:
[
p_{\text{photon}} = \frac{6.626 \times 10^{-34}}{690 \times 10^{-9}} \approx 9.61 \times 10^{-28} \, \text{kg·m/s}
]
Step 2: Equate the momentum of the ball to the momentum of the photon
Since the photon and the Ping-Pong ball have the same momentum:
[
p_{\text{ball}} = p_{\text{photon}}
]
[
m_{\text{ball}} v = p_{\text{photon}}
]
We can solve for the speed of the ball ( v ):
[
v = \frac{p_{\text{photon}}}{m_{\text{ball}}}
]
Substitute the given mass of the Ping-Pong ball (( m_{\text{ball}} = 2.60 \times 10^{-3} \, \text{kg} )):
[
v = \frac{9.61 \times 10^{-28}}{2.60 \times 10^{-3}} \approx 3.69 \times 10^{-25} \, \text{m/s}
]
Conclusion:
The speed of the Ping-Pong ball is approximately ( 3.69 \times 10^{-25} \, \text{m/s} ). This is an extremely small velocity, as the ball has much more mass than the photon, making it much slower than light in terms of speed.