Find the final velocity of the two balls if the ball with velocity v2i = -19.5 cm/s has a mass equal to half that of the ball with initial velocity v1i = +26.8 cm/s.
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You need to more fully describe the collsion
To find the final velocity of the two balls, we can use the law of conservation of momentum. According to this law, the total momentum before the collision is equal to the total momentum after the collision.
The formula for momentum is given by:
Momentum = mass * velocity
Let's denote the mass of the ball with initial velocity v1i as m1, and the mass of the ball with initial velocity v2i as m2. Given that the mass of the second ball is half that of the first ball, we have:
m2 = 1/2 * m1
Let's denote the final velocities of the balls as v1f for the first ball and v2f for the second ball.
According to the law of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision. Therefore:
m1 * v1i + m2 * v2i = m1 * v1f + m2 * v2f
Substituting the values, we have:
m1 * 26.8 cm/s + (1/2 * m1) * (-19.5 cm/s) = m1 * v1f + (1/2 * m1) * v2f
Simplifying the equation, we get:
26.8 cm/s - (1/2 * 19.5 cm/s) = v1f + (1/2 * v2f)
Now, let's solve for v1f:
v1f = 26.8 cm/s - (1/2 * 19.5 cm/s)
v1f = 26.8 cm/s - 9.75 cm/s
v1f = 17.05 cm/s
Next, let's solve for v2f:
Since we know that m2 = 1/2 * m1, we can substitute this expression into the equation:
v1f + (1/2 * v2f) = 17.05 cm/s
Simplifying the equation, we get:
v2f/2 = 17.05 cm/s - 17.05 cm/s
v2f/2 = 0 cm/s
Multiplying both sides by 2, we get:
v2f = 0 cm/s
Therefore, the final velocity of the first ball (v1f) is 17.05 cm/s, and the final velocity of the second ball (v2f) is 0 cm/s.
To find the final velocity of the two balls, we can use the principle of conservation of momentum. According to this principle, the total momentum of an isolated system remains constant before and after a collision.
The equation for conservation of momentum is:
m1 * v1i + m2 * v2i = m1 * v1f + m2 * v2f
Where:
m1 and m2 are the masses of the balls
v1i and v2i are the initial velocities of the balls
v1f and v2f are the final velocities of the balls
We are given that the mass of the ball with v2i is half that of the ball with v1i. Let's assume the mass of the ball with v1i (m1) is 2m, and the mass of the ball with v2i (m2) is m.
Applying this information to the equation, we get:
2m * 26.8 cm/s + m * (-19.5 cm/s) = 2m * v1f + m * v2f
Simplifying the equation:
53.6 m cm/s - 19.5 m cm/s = 2m * v1f + m * v2f
34.1 m cm/s = 2m * v1f + m * v2f
Now we need another equation to solve for the final velocities. This can be derived from the fact that during an elastic collision, the relative velocity of approach and separation remains the same.
So, the equation for the relative velocity is:
v1i - v2i = -(v1f - v2f)
Substituting the given values:
26.8 cm/s - (-19.5 cm/s) = -(v1f - v2f)
26.8 cm/s + 19.5 cm/s = -(v1f - v2f)
46.3 cm/s = -(v1f - v2f)
From this equation, we get:
v1f - v2f = -46.3 cm/s
Now we can solve the system of equations using substitution. Let's solve for v1f in terms of v2f:
34.1 m cm/s = 2m * v1f + m * v2f (equation 1)
v1f - v2f = -46.3 cm/s (equation 2)
Solving equation 1 for v1f:
v1f = (34.1 m - m * v2f) / (2m)
Substituting this value in equation 2:
(34.1 m - m * v2f) / (2m) - v2f = -46.3 cm/s
Simplifying the equation:
34.1 - v2f = -92.6
-v2f = -92.6 - 34.1
-v2f = -126.7
v2f = 126.7 cm/s
Now we can substitute this value of v2f back into equation 2 to solve for v1f:
v1f = -46.3 cm/s + 126.7 cm/s
v1f = 80.4 cm/s
Therefore, the final velocity of the ball with initial velocity v1i = +26.8 cm/s is v1f = 80.4 cm/s, and the final velocity of the ball with initial velocity v2i = -19.5 cm/s is v2f = 126.7 cm/s.
Note: It is important to double-check the unit conversions and use consistent units throughout the calculations to obtain accurate results.