Understanding Newton's Laws for Constant Velocity Motion

This question asks us to determine the required net external force on a particle moving with constant velocity on a frictionless surface, based on Newton's laws of motion.

Newton's First Law of Motion (Law of Inertia)

Newton's First Law states that an object will remain at rest or continue to move with a constant velocity in a straight line unless acted upon by a net external force. In this scenario, the particle is already moving with constant velocity. This implies that the conditions of Newton's First Law are met if the net external force is zero.

Newton's Second Law of Motion

Newton's Second Law provides a more quantitative relationship between force, mass, and motion. It states that the net external force ($\vec{F}_{net}$) acting on an object is equal to the product of its mass ($m$) and its acceleration ($\vec{a}$). Mathematically, this is expressed as:

$ \vec{F}_{net} = m\vec{a} $

Applying the Laws to the Scenario

The key information given is that the particle is moving with constant velocity.

• Constant Velocity Means Zero Acceleration: Velocity is a measure of both speed and direction. If the velocity is constant, it means neither the speed nor the direction is changing. Acceleration is defined as the rate of change of velocity ($\vec{a} = \frac{\Delta \vec{v}}{\Delta t}$). Since the velocity is not changing, the acceleration of the particle is zero ($\vec{a} = 0$).
• Calculating Net External Force: Using Newton's Second Law ($ \vec{F}_{net} = m\vec{a} $), if the acceleration ($\vec{a}$) is zero, the net external force must also be zero, regardless of the mass ($m$) of the particle.

$ \vec{F}_{net} = m \times 0 = 0 $

Therefore, for a particle to move with constant velocity on a frictionless horizontal surface, the net external force acting on it must be zero.

Analysis of Options

Let's examine the given options in light of this understanding:

• Option 1: States the force must be non-zero. This contradicts Newton's Second Law, as a non-zero force would cause acceleration and change the velocity.
• Option 2: Suggests a force opposite to motion. Such a force would cause deceleration, meaning the velocity would decrease, which contradicts the condition of constant velocity.
• Option 3: States the force must be zero because acceleration is zero. This aligns perfectly with our analysis derived from Newton's laws. Constant velocity implies zero acceleration, which necessitates a zero net force.
• Option 4: Relates the force magnitude to kinetic energy. Kinetic energy is $ KE = \frac{1}{2}mv^2 $. If velocity ($v$) is constant, kinetic energy is also constant. A varying force would imply varying acceleration and velocity, hence varying kinetic energy. This option is incorrect.

Conclusion on Net External Force

Based on Newton's laws, the condition of constant velocity directly implies that the net external force acting on the particle must be zero. The frictionless surface simply ensures that we only need to consider forces directly causing or opposing the change in horizontal motion, and that there's no friction force acting against the motion.