Consider a perfectly elastic collision between two objects of equal mass. Object 1 is initially moving with a velocity v = 4.33 m/s while object 2 is at rest. What are the final velocities after the collision? Enter the final velocity of object 1 first.
The velocities are switched in this equal-mass elastic case.
Object 1: 0
Object 2: 4.33 m/s
It is obvious from inspection that both total momentum and kineticenergy are conserved.
This can be proven with simultaneous momentum and energy equations, and I recommend that you try to prove it that way.
To find the final velocities after a perfectly elastic collision between two objects of equal mass, we can use the principle of conservation of linear momentum.
The principle of conservation of linear momentum states that the total momentum before the collision is equal to the total momentum after the collision.
In this case, the initial momentum is given by the mass (m) of the objects multiplied by their initial velocities (v). Since the mass and initial velocity of object 2 are both zero, its initial momentum is zero.
The initial momentum of object 1 is given by p1 = m * v1 = m * 4.33 m/s, where m is the mass of the objects.
After the collision, the final momentum of object 1 is given by p1' = m * v1', and the final momentum of object 2 is given by p2' = m * v2'.
According to the conservation of linear momentum, the total initial momentum is equal to the total final momentum, so p1 + p2 = p1' + p2'.
Since object 2 is at rest initially, its initial momentum is zero, so we have p1 = p1' + p2'.
Substituting the values, we get m * 4.33 m/s = m * v1' + m * v2'.
Since the masses of the two objects are equal (m = m), we can simplify the equation to 4.33 m/s = v1' + v2'.
Since the objects are perfectly elastic, the relative velocities of the two objects before and after the collision remain the same. In this case, since object 2 is initially at rest, its velocity after the collision will be equal to the negative of object 1's initial velocity, so v2' = -4.33 m/s.
Substituting this value back into the equation, we get 4.33 m/s = v1' + (-4.33 m/s).
Rearranging the equation, we find that v1' = 2 * 4.33 m/s.
So, the final velocity of object 1 is v1' = 8.66 m/s, and the final velocity of object 2 is v2' = -4.33 m/s.
To find the final velocities of object 1 and object 2 after a perfectly elastic collision, we can use the principles of conservation of momentum and conservation of kinetic energy.
Let's denote the final velocity of object 1 as v1' and the final velocity of object 2 as v2'.
Conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision:
m1 * v1 + m2 * v2 = m1 * v1' + m2 * v2'
In this case, since object 2 is initially at rest, its velocity (v2) is 0. Therefore, the equation simplifies to:
m1 * v1 = m1 * v1' + m2 * v2'
Since the masses of both objects are equal (m1 = m2), we can further simplify the equation to:
v1 = v1' + 0
Now, let's consider the conservation of kinetic energy. In a perfectly elastic collision, the total kinetic energy before the collision is equal to the total kinetic energy after the collision:
(1/2) * m1 * v1^2 + (1/2) * m2 * v2^2 = (1/2) * m1 * v1'^2 + (1/2) * m2 * v2'^2
Substituting the fact that v2 is 0, the equation reduces to:
(1/2) * m1 * v1^2 = (1/2) * m1 * v1'^2 + 0
Dividing both sides of the equation by (1/2) * m1, we get:
v1^2 = v1'^2
Taking the square root of both sides, we find:
v1 = v1'
This means that the final velocity of object 1 (v1') is equal to its initial velocity (v1) of 4.33 m/s.
Therefore, the final velocities after the collision are v1' = 4.33 m/s for object 1 and v2' = 0 m/s for object 2.