SPH3U: Introducing “The BIG Five” Equations of Constant Acceleration
A: Displacement for Constant Velocity and Constant Acceleration
Constant Velocity: Alice drives her car at a velocity of 20 m/s [W] for 5.0 s.
(a) Sketch her velocity-time graph, carefully labeling all values.
(b) Calculate her displacement over this time interval, using the velocity-time graph. (100 m [W])
Constant Acceleration: Bob speeds up smoothly (i.e. uniform acceleration) from 10 m/s [W] to 20 m/s [W]
over 5.0 s.
(a) Sketch his velocity-time graph, carefully labeling all values.
(b) Calculate his acceleration over this time interval, using the velocity-time graph. (2.0 m/s2 [W])
(c) Calculate his displacement over this time interval, using the velocity-time graph. (75 m [W])
B: Relating Graphs to Equations
Now consider the following graph of uniformly accelerated
motion. We will use it to find an equation to calculate the
displacement between times t1 and t2.
(a) What is the height and width of the rectangle? Write an
expression for its area using kinematic symbols.
(b) What is the height and width of the triangle? Write an expression for its area.
(c) Explain why Δv = aΔt. Use this to write a new expression for the area of the triangle that does not use
Δv.
(d) The total area represents the displacement of the object during the time interval. Write a complete
expression for the displacement (using only ∆𝑑⃗, 𝑣⃗ 1, 𝑎⃗ and Δt).
(e) Use the equation that you constructed in (d) to calculate Bob’s displacement from Part A. Use the
GRASS (Given, Required, Analyze, Solve, State) method. Your “Givens” in this example are 𝑣⃗ 1, 𝑎⃗ and
Δt, and your “Required” is ∆𝑑⃗. Compare the displacement you calculated using this equation to the
displacement you found in Part A using the velocity-time graph.
The equation you constructed in part (d) is one of the five equations of uniform (constant)
acceleration known as “The BIG Five”. These five equations relate the five kinematic variables
(∆𝑑⃗, 𝑣⃗1, 𝑣⃗2 , 𝑎⃗ and Δt) to each other. You have already encountered one other BIG five – a rearranged version of the definition of acceleration 𝑎⃗ = 𝛥𝑣⃗/𝛥𝑡. Using algebra, we can create the
other three equations.
SPH3U: The BIG Five and Solving Problems
Fill in the chart with either a checkmark (√) or an (×) to indicate whether the kinematic variable is found in
that equation. Note that all of these variables except Δt are vectors, but we are omitting the vector symbol for
clarity.
Δt
v1
v2
a
Δd
v2 = v1 + aΔt
Δd = ½(v1 + v2)Δt
Δd = v1Δt + ½aΔt2
Δd = v2Δt - ½aΔt2
(v2)2 = (v1)2 + 2aΔd
(a) How many kinematic variables does each equation have?
(b) If you wanted to use the first equation to calculate the acceleration, how many variable quantities would
you need to know?
(c) What does each of the kinematic variables mean?
v1 - initial velocity
Δt -
v2 -
Δd -
a(d) What condition must be true or approximately true to use these equations to solve problems?
Using “The BIG Five” to solve problems: As Easy as 3-4-5
As soon as you know three kinematic quantities, you can always find a fourth quantity using the
appropriate BIG Five!
Write your solutions carefully using the GRASS (Given, Required, Analyze, Solve, State)
process. Use the BIG Five chart to help you determine the appropriate equation to use.
SPH3U: “The BIG Five” – Homework Problems
For each of the following problems, (1) sketch a velocity-time graph, and then (2) use the GRASS
method and the BIG Five to answer the questions.
Problem 1. A traffic light turns green and an anxious student floors the gas pedal, causing the car to
accelerate at 3.4 m/s2 for a total of 10.0 seconds. How far did the car travel in that time? (You need
3 quantities to solve this, but it seems as if you only have 2. Hint: What is the initial velocity?)
Problem 2. An automobile safety laboratory performs crash tests of vehicles to ensure their safety
in high-speed collisions. In a head-on crash test, a Smart Car collides with a solid barrier. The car is
initially traveling at 100 km/h and then the front crumples 0.78 m during the collision until it comes
to a stop. How much time does the collision take? What was the car’s acceleration during the
collision? (Remember to convert speeds to m/s!)
(Work space for Problem 2)
Problem 3. The brakes on your car are capable of slowing down your car at a rate of 5.2 m/s2. You
are traveling at 137 km/h when you see a cop pointing a radar gun at you. What is the minimum
time in which you can get your car under the 100 km/h speed limit? (Remember to convert speeds
to m/s!)
Answers:
Problem 1: 170 m
Problem 2: 0.056 s, -4.9 × 102 m/s2
Problem 3: 2.0 s