Question

Two bodies of masses m and 4m are moving with equal kinetic energies. The ratio of their linear momenta is

• A. $1 : 2$
• B. $1 : 4$
• C. $4 : 1$
• D. $1 : 1$

Answer 1

The Correct Option is A

To determine the ratio of the linear momenta of two bodies of masses \( m \) and \( 4m \) with equal kinetic energies, we follow these steps:

Step 1: Understand the Equation for Kinetic Energy

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.

Step 2: Set Up Equations for Each Body's Kinetic Energy

Let the velocity of the mass \( m \) be \( v_1 \) and the velocity of the mass \( 4m \) be \( v_2 \). Since their kinetic energies are equal, we can equate their formulas:

\frac{1}{2} m v_1^2 = \frac{1}{2} (4m) v_2^2

Simplifying, we find:

m v_1^2 = 4m v_2^2

Cancelling the mass \( m \) from both sides, we get:

v_1^2 = 4 v_2^2

Taking the square root:

v_1 = 2 v_2

Step 3: Calculate the Linear Momentum for Both Masses

The linear momentum \( p \) of an object is given by:

p = mv

For the mass \( m \) (with velocity \( v_1 \)):

p_1 = m \cdot v_1 = m \cdot 2v_2 = 2mv_2

For the mass \( 4m \) (with velocity \( v_2 \)):

p_2 = 4m \cdot v_2 = 4mv_2

Step 4: Find the Ratio of Linear Momentum

The ratio of the momenta \( p_1 \) to \( p_2 \) is:

\frac{p_1}{p_2} = \frac{2mv_2}{4mv_2} = \frac{1}{2}

Therefore, the ratio of their linear momenta is 1 : 2.

Conclusion

The correct answer is 1 : 2, which matches the given correct answer option. Here, the reason is that the lighter body must travel faster to have the same kinetic energy, thereby affecting the linear momentum in the derived ratio.