A 4 kg mass and a 1 kg mass are moving with equal kinetic energies. The ratio of the magnitudes of their linear momenta is

Options

A. `1:2`
B. `1:1`
C. `2:1`
D. `4:1`

Answer

To find the ratio of the magnitudes of the linear momenta of a 4 kg mass and a 1 kg mass that are moving with equal kinetic energies, we can follow these steps:

### Step 1: Understand the relationship between kinetic energy and momentum
The kinetic energy (KE) of an object is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
where \(m\) is the mass and \(v\) is the velocity.

The linear momentum (p) is given by:
\[ p = mv \]

### Step 2: Set up the equations for both masses
Let the kinetic energies of both masses be equal. Therefore, we can write:
\[ KE_1 = KE_2 \]
For the 4 kg mass:
\[ KE_1 = \frac{1}{2} (4) v_1^2 \]
For the 1 kg mass:
\[ KE_2 = \frac{1}{2} (1) v_2^2 \]

### Step 3: Equate the kinetic energies
Setting the two kinetic energies equal gives us:
\[ \frac{1}{2} (4) v_1^2 = \frac{1}{2} (1) v_2^2 \]
This simplifies to:
\[ 4 v_1^2 = v_2^2 \]

### Step 4: Solve for the velocities
From the equation \(4 v_1^2 = v_2^2\), we can express \(v_2\) in terms of \(v_1\):
\[ v_2 = \sqrt{4} v_1 = 2 v_1 \]

### Step 5: Calculate the momenta
Now we can calculate the momenta for both masses:
For the 4 kg mass:
\[ p_1 = 4 v_1 \]
For the 1 kg mass:
\[ p_2 = 1 v_2 = 1 (2 v_1) = 2 v_1 \]

### Step 6: Find the ratio of the momenta
Now we can find the ratio of the magnitudes of their linear momenta:
\[ \frac{p_1}{p_2} = \frac{4 v_1}{2 v_1} = \frac{4}{2} = 2 \]

Thus, the ratio of the magnitudes of their linear momenta is:
\[ \frac{p_1}{p_2} = 2:1 \]

### Final Answer
The ratio of the magnitudes of their linear momenta is \(2:1\).

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