The two particles are both moving to the right. Particle 1 catches up with particle 2 and collides with it. The particles stick together and continue on with velocity vf. Which of these statements is true?
A. vf=v2. B. vf is less than v2. C. vf is greater than v2, but less than v1. D. vf=v1. E. vf is greater than v1.
To determine the relationship between the final velocity (vf) and the initial velocities (v1 and v2) of the two particles after collision, we need to apply the principle of conservation of linear momentum.
The conservation of linear momentum states that the total momentum of a system remains constant if no external forces act on it. In this case, since the two particles stick together after collision, we can consider them as a system with no external forces.
The linear momentum (p) of an object is given by the product of its mass (m) and velocity (v). Mathematically, this can be represented as p = mv.
Before the collision, the linear momentum of the first particle (particle 1) is p1 = m1 * v1, and the linear momentum of the second particle (particle 2) is p2 = m2 * v2.
Since the particles stick together after the collision, we can combine their masses and velocities as a single object. The combined mass is the sum of the individual masses, m = m1 + m2.
The final velocity of the particles (vf) after collision can be obtained by dividing the total momentum after collision (pf) by the combined mass (m). Mathematically, this can be expressed as vf = pf / m.
Using the principle of conservation of linear momentum, we know that the total momentum before collision (p1 + p2) is equal to the total momentum after collision (pf). Therefore, we have p1 + p2 = pf.
Substituting the expressions for momentum, we get m1 * v1 + m2 * v2 = (m1 + m2) * vf.
Rearranging the equation, we get vf = (m1 * v1 + m2 * v2) / (m1 + m2).
From this equation, we can determine the relationship between vf, v1, and v2.
Comparing the equation to the given options:
A. vf = v2. This statement is not always true. The final velocity may or may not be equal to v2, depending on the masses of the particles.
B. vf is less than v2. This statement is not always true. The final velocity may or may not be less than v2, depending on the masses of the particles.
C. vf is greater than v2, but less than v1. This statement is not always true. The final velocity may or may not be in this range, depending on the masses and initial velocities of the particles.
D. vf = v1. This statement is not always true. The final velocity may or may not be equal to v1, depending on the masses of the particles.
E. vf is greater than v1. This statement is always true. The final velocity of the particles after collision will be greater than the initial velocity of particle 1, regardless of the masses involved.
Therefore, the correct answer is E. vf is greater than v1.