One ball has 7 times the mass and twice the speed of another.
1. How does the momentum of the more massive ball compare to the momentum of the less massive one? Find the ratio of the momenta.
2. How does the kinetic energy of the more massive ball compare to the kinetic energy of the less massive one? Find the ratio of the kinetic energies
Step 1: Calculate the momentum of each ball:
For the more massive ball:
Let the mass of the less massive ball be m.
The mass of the more massive ball is 7m.
The velocity of the less massive ball is v.
The velocity of the more massive ball is 2v.
The momentum of an object is given by the product of its mass and velocity.
Therefore, the momentum of the less massive ball is m * v.
And the momentum of the more massive ball is (7m) * (2v).
Step 2: Compare the two momenta:
The ratio of the momenta is given by:
Momentum of the more massive ball / Momentum of the less massive ball.
Substituting the values, the ratio of the momenta is:
[(7m) * (2v)] / [m * v]
Simplifying further, the ratio of the momenta is:
(14mv) / (mv)
Step 3: Simplify the equation:
The mass and velocity are common in the numerator and denominator, so they can be canceled out.
The simplified ratio of the momenta is:
14/1, which is equal to 14.
Therefore, the momentum of the more massive ball is 14 times greater than the momentum of the less massive ball.
Moving on to part 2:
Step 4: Calculate the kinetic energy of each ball:
The kinetic energy of an object is given by the formula:
Kinetic energy = (1/2) * mass * (velocity)^2.
For the less massive ball, the kinetic energy is:
(1/2) * m * v^2.
For the more massive ball, the kinetic energy is:
(1/2) * 7m * (2v)^2.
Step 5: Compare the two kinetic energies:
The ratio of the kinetic energies is given by:
Kinetic energy of the more massive ball / Kinetic energy of the less massive ball.
Substituting the values, the ratio of the kinetic energies is:
[(1/2) * 7m * (2v)^2] / [(1/2) * m * v^2]
Simplifying further, the ratio of the kinetic energies is:
[(7m * 4v^2) / (m * v^2)]
Step 6: Simplify the equation:
The mass and velocity terms are common in the numerator and denominator, so they can be canceled out.
The simplified ratio of the kinetic energies is:
7 * 4, which is equal to 28.
Therefore, the kinetic energy of the more massive ball is 28 times greater than the kinetic energy of the less massive ball.
To find the ratio of the momenta and kinetic energies, we need to understand the equations that relate these quantities.
1. The momentum (p) of an object is given by the product of its mass (m) and velocity (v): p = m * v. Therefore, if one ball has 7 times the mass (m2 = 7 * m1) and twice the speed (v2 = 2 * v1) of the other ball, we can find the ratio of their momenta as follows:
p2 = (m2 * v2) = (7 * m1) * (2 * v1) = 14 * (m1 * v1) = 14 * p1
So, the momentum of the more massive ball is 14 times greater than the momentum of the less massive ball. The ratio of the momenta is therefore 14:1.
2. The kinetic energy (K) of an object is given by the equation K = (1/2) * m * v^2. By substituting the given values into this equation, we can find the ratio of their kinetic energies:
K2 = (1/2) * (m2 * v2^2) = (1/2) * (7 * m1) * (2 * v1)^2 = 7 * 2^2 * (1/2) * (m1 * v1^2) = 7 * 2^2 * K1 = 28 * K1
Therefore, the kinetic energy of the more massive ball is 28 times greater than the kinetic energy of the less massive ball. The ratio of the kinetic energies is 28:1.
7 m * 2 v = 14 m v
(1/2)(7m)(2v)^2 = (1/2)(7)(4) mv^2
= 28 * (1/2)m v^2