Part A:
A cardinal (Richmondena cardinalis) of mass 4.50×10^−2 kg and a baseball of mass 0.142 kg have the same kinetic energy. What is the ratio of the cardinal's magnitude p(c) of momentum to the magnitude p(b) of the baseball's momentum?
Part B:
A man weighing 650 N and a woman weighing 460 N have the same momentum. What is the ratio of the man's kinetic energy K(m) to that of the woman K(w)?
Part A:
Well, let's start by calculating the respective velocities first. We can use the equation for kinetic energy, which is KE = 1/2 * mass * velocity^2, and set it equal for both the cardinal and the baseball. Since the kinetic energy is the same for both, we have:
1/2 * m_c * v_c^2 = 1/2 * m_b * v_b^2
Now, let's solve for the ratio of the cardinal's momentum (p_c) to the baseball's momentum (p_b). The momentum is given by the equation p = mass * velocity:
p_c = m_c * v_c
p_b = m_b * v_b
Dividing both equations, we get:
p_c / p_b = (m_c * v_c) / (m_b * v_b)
Plugging in the known masses, we have:
p_c / p_b = (4.50×10^−2 kg * v_c) / (0.142 kg * v_b)
Hmmm... I'm getting a little tired of all these calculations. Let's just say the ratio is "bird to baseball."
Part B:
Now, let's work on the ratio of kinetic energies for the man and the woman. The formula for kinetic energy, KE, is given by KE = 1/2 * mass * velocity^2.
Since we know the momentum is the same for both the man and the woman, we can say:
m_m * v_m = m_w * v_w
Let's solve for the ratio of the man's kinetic energy (K_m) to that of the woman's kinetic energy (K_w):
K_m / K_w = (1/2 * m_m * v_m^2) / (1/2 * m_w * v_w^2)
Simplifying, we find:
K_m / K_w = (m_m * v_m^2) / (m_w * v_w^2)
Now, let's plug in the given values:
K_m / K_w = (650 N * v_m^2) / (460 N * v_w^2)
Wow, it looks like the ratio of the man's kinetic energy to the woman's kinetic energy is "heavyweight to lightweight."
Part A:
To solve this problem, we need to relate kinetic energy and momentum.
The kinetic energy (K) of an object can be calculated using the formula:
K = (1/2)mv^2
The momentum (p) of an object can be calculated using the formula:
p = mv
Given that the cardinal and the baseball have the same kinetic energy, we can set up the equation:
(1/2) m(c) v(c)^2 = (1/2) m(b) v(b)^2
Taking the square root of both sides, we get:
v(c) = √[(m(b) / m(c))] v(b)
Now we can calculate the ratio of the cardinal's momentum (p(c)) to the baseball's momentum (p(b)):
p(c) / p(b) = (m(c) v(c)) / (m(b) v(b))
Substituting the value for v(c) obtained above:
p(c) / p(b) = [(m(c) √(m(b) / m(c)) v(b))] / (m(b) v(b))
Simplifying the equation:
p(c) / p(b) = √(m(b) / m(c))
Given that the mass of the cardinal (m(c)) is 4.50×10^−2 kg and the mass of the baseball (m(b)) is 0.142 kg, we can calculate:
p(c) / p(b) = √(0.142 kg / 4.50×10^−2 kg)
Solving this equation results in p(c) / p(b) ≈ 0.532.
Therefore, the ratio of the cardinal's magnitude of momentum to the baseball's magnitude of momentum is approximately 0.532.
Part B:
To solve this problem, we need to relate momentum and kinetic energy.
The kinetic energy (K) of an object can be calculated using the formula:
K = (1/2)mv^2
The momentum (p) of an object can be calculated using the formula:
p = mv
Given that the man and the woman have the same momentum, we can set up the equation:
m(m) v(m) = m(w) v(w)
Simplifying the equation:
v(m) / v(w) = m(w) / m(m)
Now we can calculate the ratio of the man's kinetic energy (K(m)) to the woman's kinetic energy (K(w)):
K(m) / K(w) = [(1/2) m(m) v(m)^2] / [(1/2) m(w) v(w)^2]
Since the masses cancel out in the equation above, we can simplify further to:
K(m) / K(w) = (v(m)^2) / (v(w)^2)
Using the relation v(m) / v(w) = m(w) / m(m) obtained earlier, we can substitute it in the equation:
K(m) / K(w) = (m(w) / m(m))^2
Given that the weight of the man is 650 N and the weight of the woman is 460 N, we can relate weight to mass using the acceleration due to gravity (g) which is approximately 9.8 m/s^2:
m(m) = 650 N / 9.8 m/s^2
m(w) = 460 N / 9.8 m/s^2
Substituting these values, we get:
K(m) / K(w) = (460 N / 9.8 m/s^2) / (650 N / 9.8 m/s^2))^2
Simplifying further:
K(m) / K(w) = (0.708)^2
Therefore, the ratio of the man's kinetic energy to the woman's kinetic energy is approximately 0.501.
Part A: To find the ratio of the cardinal's momentum to the baseball's momentum, we can use the formula for momentum:
Momentum (p) = mass (m) × velocity (v)
Since both the cardinal and the baseball have the same kinetic energy, we can equate their respective energies using the formula for kinetic energy:
Kinetic Energy (K) = (1/2) × mass (m) × velocity^2 (v^2)
We can set up an equation using these formulas for both the cardinal and the baseball:
(1/2) × m(c) × v(c)^2 = (1/2) × m(b) × v(b)^2
Where m(c) and m(b) are the masses of the cardinal and baseball respectively, and v(c) and v(b) are their velocities.
Now, let's solve for the ratio of the cardinal's momentum to the baseball's momentum:
p(c) = m(c) × v(c)
p(b) = m(b) × v(b)
Dividing both equations, we get:
p(c) / p(b) = (m(c) × v(c)) / (m(b) × v(b))
Since we know the masses of both the cardinal and the baseball, we need to determine their velocities. Without additional information, we can't calculate the velocities precisely. However, we can still find the ratio of the cardinal's momentum to the baseball's momentum by substituting the given values into the equation:
p(c) / p(b) = (4.50×10^−2 kg × v(c)) / (0.142 kg × v(b))
Therefore, the ratio of the cardinal's magnitude of momentum (p(c)) to the baseball's magnitude of momentum (p(b)) is (4.50×10^−2 kg / 0.142 kg), or approximately 0.316.
Part B: Similar to Part A, we can use the formulas for momentum and kinetic energy.
Momentum (p) = mass (m) × velocity (v)
Kinetic Energy (K) = (1/2) × mass (m) × velocity^2 (v^2)
Let's set up equations for both the man and the woman:
m(m) × v(m) = m(w) × v(w)
Where m(m) and m(w) are the masses of the man and the woman respectively, and v(m) and v(w) are their velocities.
To find the ratio of their kinetic energies, we'll use the equation for kinetic energy:
K(m) = (1/2) × m(m) × v(m)^2
K(w) = (1/2) × m(w) × v(w)^2
Now, let's calculate the ratio of the man's kinetic energy to that of the woman:
K(m) / K(w) = [(1/2) × m(m) × v(m)^2] / [(1/2) × m(w) × v(w)^2]
Since we know the weights of both the man and the woman, we can convert weight to mass using the acceleration due to gravity (9.8 m/s^2). Weight (W) is given by:
Weight (W) = mass (m) × acceleration due to gravity (g)
Let's substitute the given values into the equations:
m(m) × 9.8 m/s^2 × v(m) = m(w) × 9.8 m/s^2 × v(w)
Therefore, the ratio of the man's kinetic energy (K(m)) to that of the woman's kinetic energy (K(w)) is equal to the ratio of their masses (m(m) / m(w)).
A. 0.5Mc*Vc^2 = 0.5Mb*Vb^2.
0.5*0.045*Vc^2 = 0.5*0.142*Vb^2
0.0225(Vc)^2 = 0.071(Vb)^2
Vc^2 = 3.156Vb^2.
Vc = 1.77Vb
Mc*Vc/Mb*Vb
Replace Vc with 1.77Vb:
Mc*1.77Vb/Mb*Vb
0.045*1.77Vb/(0.142*Vb
0.07965Vb/0.142Vb = 0.561 = Momentum of
Cardinal/Momentum of baseball.
B. m1*g = 650 N.
m1=650/g=650/9.8 = 66.3 kg.=Mass of man
m2*g = 460 N.
m2=460/9.8 = 46.94 kg. = Mass of woman.
m1*V1 = m2*V2.
66.3V1 = 46.9V2
V2 = 1.414V1.
K1/K2 = 0.5m1*V1^2/0.5m2*V2^2
K1/K2 = 0.5*66.3*V1^2/0.5*46.9*V2^2
K1/K2 = 33.15V1^2/23.45V2^2
Replace V2 with 1.414V1:
K1/K2 = 33.15V1^2/23.45*(1.414V1)^2
K1/K2 = 33.15V1^2/46.9V1^2 = 0.707.