The Key Equation
The most fundamental equation for linear motion with constant acceleration is:
* v = u + at
* v: Final velocity
* u: Initial velocity
* a: Acceleration
* t: Time
Derivation and Other Equations
This equation is derived from the definition of acceleration (a = Δv/Δt) and assuming constant acceleration. From it, we can derive other useful equations:
* s = ut + ½at² (Displacement)
* v² = u² + 2as (Relationship between velocities and displacement)
Why These Equations Apply Only to Acceleration
* Constant Acceleration: The equations above are valid only when the acceleration is constant. If acceleration is changing, we need more complex calculus-based methods.
* Zero Acceleration (Constant Velocity): If the acceleration is zero (meaning the object is moving at a constant velocity), the equations simplify significantly. For example, the first equation becomes v = u, meaning the final velocity is equal to the initial velocity.
Important Considerations
* Direction: These equations are vector equations. That means you need to be mindful of the direction of the acceleration, velocity, and displacement.
* Sign Convention: Be consistent with your sign convention (e.g., positive for motion to the right, negative for motion to the left).
Example
Let's say a car starts from rest (u = 0 m/s) and accelerates at 2 m/s² for 5 seconds. We can use the equations to find:
* Final velocity (v): v = 0 + (2 m/s²)(5 s) = 10 m/s
* Displacement (s): s = (0 m/s)(5 s) + ½(2 m/s²)(5 s)² = 25 m
In summary, these equations are vital for describing linear motion when an object is undergoing a constant change in velocity. They are the building blocks for understanding more complex motion.
