Technology
Why You Cant Use vx/t and av/t for Uniformly Accelerated Motion
Why You Can't Use vx/t and av/t for Uniformly Accelerated Motion
In physics, when dealing with uniformly accelerated motion, simple equations like v x/t and a v/t can lead to misunderstandings unless they are used in the right contexts. This article will explore the limitations of these equations, discuss why they are not suitable for uniformly accelerated motion, and provide the correct mathematical tools for solving such motion problems.
1. Uniformly Accelerated Motion: A Constant Acceleration Phenomenon
Uniformly accelerated motion (also known as linear motion) refers to the case where the acceleration a is constant over time. In such situations, the velocity v changes over time, requiring the use of average or specific values rather than instantaneous ones.
Why v x/t fails:
The equation v x/t assumes a constant velocity, whereas in uniformly accelerated motion, velocity changes over time. This means that the velocity at any given moment is not equal to the average velocity over the entire process. Instead, the average velocity should be used, given by the formula v_{avg} (v_i v_f) / 2, where v_i is the initial velocity and v_f is the final velocity.
Illustration: Imagine a car accelerating from 0 to 20 m/s in 5 seconds. The simple equation v x/t would suggest the car's velocity is constantly 4 m/s, which is clearly incorrect.
2. Average vs. Instantaneous Values
It is crucial to distinguish between instantaneous velocity and average velocity in uniformly accelerated motion. The instantaneous velocity at any point in time is the velocity at that exact moment, whereas the average velocity is the overall velocity over a specific period.
Why a v/t fails:
The equation a v/t is only valid for situations where velocity and acceleration are directly proportional and constant. In uniform acceleration, the acceleration is constant but the relationship with velocity is not linear. Therefore, using a v/t would not accurately represent the constant acceleration in uniformly accelerated motion.
Correct approach: The correct formula for acceleration is a Δv / Δt, which means acceleration is the change in velocity over the change in time.
3. Kinematic Equations for Uniformly Accelerated Motion
To accurately describe uniformly accelerated motion, a set of kinematic equations is necessary. These equations account for constant acceleration and can be used to relate displacement, initial and final velocities, acceleration, and time. The key kinematic equations for uniformly accelerated motion are:
v_f v_i at: This equation describes the final velocity based on the initial velocity, acceleration, and time. x v_i t (1/2)at^2: This equation defines the displacement based on the initial velocity, acceleration, and time. v_f^2 v_i^2 2ax: This equation relates the final velocity, initial velocity, acceleration, and displacement.These equations are particularly useful because they account for the changes in velocity due to constant acceleration, making them indispensable in problem-solving.
4. Conclusion
In summary, while the equations v x/t and a v/t can be useful in specific scenarios, such as constant velocity situations, they do not accurately describe the relationships present in uniformly accelerated motion. To solve problems involving constant acceleration, it is essential to use the appropriate kinematic equations that incorporate the correct definitions of velocity, acceleration, and displacement.