A proton is traveling to the right at 231 km/s. It has a head-on perfectly elastic collision with a carbon atom. The mass of the carbon atom is 12 times the mass of the proton, what is the speed of proton after the collision?
To solve this problem, we can use the concept of conservation of momentum and kinetic energy.
1. Let's assume the initial velocity of the proton is v1 (to the right) and the mass of the proton is m1.
- v1 = 231 km/s (given)
- m1 = mass of proton
2. The mass of the carbon atom is 12 times the mass of the proton.
- m2 = 12 * m1
3. Since the collision is perfectly elastic, both momentum and kinetic energy are conserved.
- Conservation of momentum: m1 * v1 = m1 * v1' + m2 * v2'
- Conservation of kinetic energy: (1/2) * m1 * v1^2 = (1/2) * m1 * v1'^2 + (1/2) * m2 * v2'^2
4. We can re-arrange the equations to solve for v1' and v2'.
- v1' = (m1 - m2) / (m1 + m2) * v1
- v2' = (2 * m1) / (m1 + m2) * v1
5. Substituting the given values:
- v1' = (m1 - m2) / (m1 + m2) * 231 km/s
- v2' = (2 * m1) / (m1 + m2) * 231 km/s
6. Since we want to find the speed of the proton after the collision, we can substitute m2 using the relation m2 = 12 * m1.
- v1' = (m1 - 12 * m1) / (m1 + 12 * m1) * 231 km/s
- v2' = (2 * m1) / (m1 + 12 * m1) * 231 km/s
7. Simplifying the expressions:
- v1' = -11/13 * 231 km/s
- v2' = 2/13 * 231 km/s
8. Finally, we can calculate the speed of the proton after the collision using the magnitude of the velocity:
- Speed of proton after collision = |v1'| = |-11/13 * 231 km/s|
9. Calculating the value:
- Speed of proton after collision = 11/13 * 231 km/s
- Speed of proton after collision = 198 km/s (approx.)
Therefore, the speed of the proton after the collision is approximately 198 km/s.
To determine the speed of the proton after the collision, we can use the principles of conservation of momentum and conservation of kinetic energy.
Conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision. Momentum is calculated by multiplying mass and velocity.
Let's call the mass of the proton as mp and the mass of the carbon atom as mc. Given that the mass of the carbon atom is 12 times the mass of the proton (mc = 12mp), we can say that the total initial momentum before the collision is:
P_initial = (mp × vp) + (mc × vc)
Where vp is the initial velocity of the proton (231 km/s) and vc is the initial velocity of the carbon atom.
During a perfectly elastic head-on collision, both momentum and kinetic energy are conserved. So, the total final momentum after the collision is:
P_final = (mp × v'p) + (mc × v'c)
Where v'p is the final velocity of the proton and v'c is the final velocity of the carbon atom.
Since the collision is head-on and perfectly elastic, the carbon atom is at rest initially (vc = 0). Therefore, our equation for initial momentum simplifies to:
P_initial = mp × vp
Applying the principle of conservation of momentum:
P_initial = P_final
mp × vp = (mp × v'p) + (mc × v'c)
Now, we need to express the final velocities in terms of the proton's velocity after the collision (v'p).
When two objects undergo an elastic collision, the relative velocities before and after the collision remain the same. The relationship between the velocities after the collision and the initial velocities can be expressed as:
v'p = [(mp - mc) / (mp + mc)] × vp
Now substitute this relationship for v'p in the momentum conservation equation:
mp × vp = (mp × [(mp - mc) / (mp + mc)] × vp) + (mc × v'c)
In the case of a perfectly elastic collision, the kinetic energy is also conserved. The kinetic energy before the collision is equal to the kinetic energy after the collision. The kinetic energy is calculated using the formula:
KE = 0.5 × mass × velocity^2
Initially, the kinetic energy is:
KE_initial = 0.5 × mp × vp^2
And after the collision, the kinetic energy is:
KE_final = 0.5 × mp × v'p^2 + 0.5 × mc × v'c^2
Since the kinetic energy is conserved:
KE_initial = KE_final
Substituting the values and the expression of v'p from the momentum conservation equation:
0.5 × mp × vp^2 = 0.5 × mp × [(mp - mc) / (mp + mc)]^2 × vp^2 + 0.5 × mc × v'c^2
Now, we can solve this equation to find v'c, the final velocity of the carbon atom. Once we have v'c, we can substitute it back into the momentum conservation equation to find v'p, the final velocity of the proton.
By following these steps and performing the calculations, you can determine the speed of the proton after the collision.