Understanding the Effect of Doubling Velocity

The question asks how certain properties of a body change when its velocity is doubled. We need to examine the definitions and formulas for momentum, kinetic energy, acceleration, and potential energy to determine the effect of a change in velocity.

Relationship between Velocity and Momentum

Momentum ($\vec{p}$) is a vector quantity defined as the product of a body's mass ($m$) and its velocity ($\vec{v}$). The formula for momentum is:

$\vec{p} = m \vec{v}$

Since mass ($m$) is a scalar and is constant for a given body, the magnitude of momentum ($p$) is directly proportional to the magnitude of velocity ($v$):

$p = m v$

If the initial velocity is $v_1$ and the initial momentum is $p_1 = m v_1$, then if the velocity is doubled to $v_2 = 2v_1$, the new momentum $p_2$ will be:

$p_2 = m v_2 = m (2v_1) = 2 (m v_1) = 2 p_1$

Therefore, when the velocity of a body is doubled, its momentum is also doubled.

Relationship between Velocity and Kinetic Energy

Kinetic energy (KE) is the energy a body possesses due to its motion. It is a scalar quantity defined by the formula:

$\text{KE} = \frac{1}{2} m v^2$

Where $m$ is the mass and $v$ is the magnitude of velocity. Kinetic energy is proportional to the square of the velocity ($v^2$).

If the initial velocity is $v_1$ and the initial kinetic energy is $\text{KE}_1 = \frac{1}{2} m v_1^2$, then if the velocity is doubled to $v_2 = 2v_1$, the new kinetic energy $\text{KE}_2$ will be:

$\text{KE}_2 = \frac{1}{2} m v_2^2 = \frac{1}{2} m (2v_1)^2 = \frac{1}{2} m (4v_1^2) = 4 \left(\frac{1}{2} m v_1^2\right) = 4 \text{KE}_1$

Therefore, when the velocity of a body is doubled, its kinetic energy becomes four times its original value (it is quadrupled), not doubled.

Examining Other Properties

Acceleration: Acceleration is the rate of change of velocity. Doubling the instantaneous velocity does not directly imply that the acceleration is doubled. Acceleration is typically caused by a net force ($\vec{a} = \vec{F}/m$). Unless the force causing the motion changes in a specific way related to velocity, doubling velocity doesn't necessarily double acceleration.

Potential Energy: Potential energy (like gravitational or elastic potential energy) depends on the position or configuration of a body in a force field, not directly on its velocity.

Analyzing the Options

• Option 1: Momentum is double - As shown by the formula $\vec{p} = m \vec{v}$, momentum is directly proportional to velocity. Doubling velocity doubles momentum. This statement is correct.
• Option 2: Kinetic energy is doubled - As shown by the formula $\text{KE} = \frac{1}{2} m v^2$, kinetic energy is proportional to the square of velocity. Doubling velocity quadruples kinetic energy. This statement is incorrect.
• Option 3: Acceleration is doubled - Doubling the instantaneous velocity does not imply that the acceleration is doubled. This statement is incorrect in general.
• Option 4: Potential energy is doubled - Potential energy depends on position, not velocity. This statement is incorrect.

Based on the analysis, when the velocity of a body is doubled, only its momentum is doubled among the given options.