Determining the Final Velocity After Force Application

In the realm of introductory physics, understanding the relationships between force, mass, and velocity is crucial. This article delves into a specific example, where we explore how a force affects the velocity of a moving object. Specifically, we'll analyze a scenario in which a 1 kg mass moving northward at 30 m/s experiences a 10 N force due east for 4 seconds.

Understanding the Problem

The problem at hand involves a body of 1 kg mass initially moving at 30 m/s due north. It encounters a force of 10 N due east for 4 seconds. Our goal is to determine the final velocity of the body after the force ceases to act.

Step-by-Step Solution

To solve this problem, we shall follow a systematic approach. Let's break it down into several stages: Initial Velocity Vector:

The initial velocity (vec{v_i}) can be represented as:

[vec{v_i} 30 , text{m/s} , text{(North)}] Acceleration due to the Force:

Using Newton's second law, (F ma), we can calculate the acceleration (vec{a}) in the eastward direction:

[vec{a} frac{vec{F}}{m} frac{10 , text{N}}{1 , text{kg}} 10 , text{m/s}^2 , text{(East)}] Change in Velocity:

Since the force acts for 4 seconds, the change in velocity (Delta vec{v}) can be calculated as:

[Delta vec{v} vec{a} cdot t 10 , text{m/s}^2 cdot 4 , text{s} 40 , text{m/s} , text{(East)}] Final Velocity Vector:

The final velocity (vec{v_f}) is the sum of the initial velocity and the change in velocity:

[vec{v_f} vec{v_i} Delta vec{v} (0 , text{m/s}, 30 , text{m/s}) (40 , text{m/s}, 0 , text{m/s}) (40 , text{m/s}, 30 , text{m/s})] Magnitude of the Final Velocity:

The magnitude of the final velocity (v_f) can be calculated using the Pythagorean theorem:

[v_f sqrt{40^2 30^2} sqrt{1600 900} sqrt{2500} 50 , text{m/s}] Direction of the Final Velocity:

The direction (theta) can be found using the tangent function:

[theta tan^{-1}left(frac{30}{40}right) tan^{-1}left(frac{3}{4}right) approx 36.87^circ , text{North of East}]

Final Result

After the force ceases to act, the velocity of the body is approximately 50 m/s on a course of 053 (approximately northeast). To summarize, the information can be compactly represented as follows: Magnitude of Velocity: 50 m/s Direction: 36.87deg; North of East

Additional Insight

By drawing this scenario as a triangle, we can visualize a 3-4-5 right triangle, confirming our calculations. The 30 m/s (north) and 40 m/s (east) components form the legs of this right triangle, while the hypotenuse represents the final velocity (50 m/s) at an angle of 36.87deg; north of east.

Conclusion

Understanding the effects of forces on velocity in physics is fundamental for solving a wide array of problem scenarios. This example illustrates how to determine the final velocity of an object given its initial state and the application of a force. Employing vector addition and the Pythagorean theorem provides a powerful method for resolving such problems accurately. By mastering these concepts, students and professionals alike can confidently tackle more complex physics problems involving motion and forces.