The BIG Five
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SPH3U: Deriving The BIG Five Equations of Uniform Acceleration
Each of the BIG Five equations of uniform (constant) acceleration can be derived from just 2 things:
1. The basic equation for 2. A general v t graph with initial velocity v1 and final velocity v2 . acceleration, where v1 is (Remember, the straight line indicates constant acceleration, and the area under the curve is the displacement.) the initial velocity, and v2 is the final velocity. v 2 v v v a 2 1 v t t 1
t
In this exercise, you will work together to derive each of the BIG Five, using the clues given. NOTE: You will NOT be required to derive these equations on a test or exam, but it is important to know where they come from when you apply them to a problem.
Equation #1: Rearrange the basic equation of acceleration to obtain equation #1: v2 = v1 + at
Equation #2:
Use the velocity-time graph to find the displacement, expressed as equation #2: d = ½(v1 + v2)t Hint: find the area under the curve in the velocity-time graph as the sum of the area of a rectangle (area = l×w) and the area of a triangle (area = ½ b × h), and then rearrange to get the equation. Equation #3: Use the velocity-time graph and the equation for acceleration to find the displacement, expressed as 2 equation #3: Δd = v1t + ½at Hint: Start by finding the area under the curve in the velocity-time graph as the sum of the area of a rectangle and triangle, as in equation #2, but this time, use the fact that v2 v1 at , and substitute it into your equation.
Equation #4: Use the velocity-time graph and the equation for acceleration to find the displacement, expressed as 2 equation #4: Δd = v2t - ½at Hint: Start by finding the area under the curve in the velocity-time graph as the area of a large rectangle and subtract the area of a triangle. Use v2 v1 at , and substitute it into your equation.
Extra Challenge: Equation #5: 2 2 See if you can derive equation #5: vf = vi + 2aΔd, where vf is the final velocity and vi is the initial velocity.