Resolving the Application of Force to Reverse Motion

Understanding how forces affect the motion of objects is a fundamental concept in physics. In this article, we explore a specific scenario where a 0.40 kg object on a frictionless surface is moving with a velocity of 30 m/s, and a force of 2.0 N is applied to reverse its direction. We will use the impulse-momentum theorem and Newton's second law to determine the duration for which the force is applied.

Impulse-Momentum Theorem

The impulse-momentum theorem is a key principle in physics that connects the impulse on an object to the change in its momentum. This theorem is expressed as:

Impulse Change in Momentum

F Delta;t m Delta;v

Where:

F is the force applied (2.0 N) Delta;t is the time interval for which the force is applied m is the mass of the object (0.40 kg) Delta;v is the change in velocity

In this problem, the velocity of the object is reversed, so the change in velocity is:

Delta;v final velocity - initial velocity -30 m/s - 30 m/s -60 m/s

Substituting the values into the impulse-momentum equation:

2.0 N Delta;t 0.40 kg times; -60 m/s

Solving for Delta;t:

Delta;t (0.40 kg times; -60 m/s) / 2.0 N -12 s

A negative sign indicates that the force was applied in the opposite direction of the initial velocity. Therefore, the force was applied for 12 seconds.

Newton's Second Law of Motion

Newton's second law of motion states that the force applied to an object is equal to the mass of the object times its acceleration (F ma). In this scenario, the force is being used to reverse the velocity:

F ma 0.40 kg × a

Since acceleration is the rate of change of velocity (a Delta;v / Delta;t), we can substitute the change in velocity:

a -60 m/s / Delta;t

Solving for Delta;t:

Delta;t -60 m/s / (2.0 N / 0.40 kg) -60 m/s / 5 m/s2

Delta;t 12 s

The negative sign in the acceleration indicates that the force is acting in the opposite direction of the initial motion. Therefore, the force was applied for 12 seconds to reverse the velocity.

Revisiting and Clarifying Concepts

It is essential to note that the application of force to reverse the motion must consider the direction of the force and its consistency with the change in velocity and momentum. In the given problem, the force of -2.0 N is applied to reverse the motion, and the time is determined as follows:

Using the relationship:

F ma

Acceleration (a) -2.0 N / 0.40 kg -5 m/s2

Now, using:

v u at

Where final velocity (v) -30 m/s, initial velocity (u) 30 m/s, and acceleration (a) -5 m/s2.

-30 m/s 30 m/s (-5 m/s2)t

-60 m/s -5 m/s2

t 12 s

The force and time for reversing the velocity are consistently determined as 12 seconds, and the negative sign in the acceleration indicates the direction of the force.

Conclusion

By applying the principles of the impulse-momentum theorem and Newton's second law of motion, we have determined that the force of 2.0 N was applied for 12 seconds to reverse the velocity of a 0.40 kg object. This exercise highlights the importance of consistency in the direction of forces and the changes in velocity and momentum.