Equations of motion, finding the equations of motion, Grade 10 Physics
Equations of motion

In this chapter we will look at the third way to describe motion. We have looked at describing motion in terms of graphs and words. In this section we examine equations that can be used to describe motion. This section is about solving problems relating to uniformly accelerated motion. In other words, motion at constant acceleration. The following are the variables that will be used in this section:

• v$_{i}$ = initial velocity (m·s$^{−1}$) at t = 0 s
• v$_{f}$ = final velocity (m·s$^{−1}$) at time t
• ∆x = displacement (m)
• t = time (s)
• ∆t = time interval (s)
• a = acceleration (m·s$^{−2}$)

The questions can vary a lot, but the following method for answering them will always work. Use this when attempting a question that involves motion with constant acceleration. You need any three known quantities (v$_{i}$, v$_{f}$ , ∆x, t or a) to be able to calculate the fourth one.

v$_{f}$ = v$_{i}$ + at (3.1)
Δ x = \[\dfrac{v_i + v_f}{2}\]t (3.2)
Δ x = v$_{i}$t + \[\dfrac{1}{2}\]at² (3.3)
v$_{f}$² = v$_{i}$² + 2a. Δx (3.4)​

1. Read the question carefully to identify the quantities that are given. Write them down.
2. Identify the equation to use. Write it down!!!
3. Ensure that all the values are in the correct unit and fill them in your equation.
4. Calculate the answer and fill in its unit.

Finding the Equations of Motion
The following does not form part of the syllabus and can be considered additional information.
Derivation of Equation 3.1
According to the definition of acceleration:

\[a = \dfrac{\Delta v}{\Delta t}\]​

where ∆v is the change in velocity, i.e. ∆v = v$_{f}$ - v$_{i}$. Thus we have

v$_{f}$ = v$_{i}$ + at​

Derivation of Equation 3.2
We have seen that displacement can be calculated from the area under a velocity vs. time graph. For uniformly accelerated motion the most complicated velocity vs. time graph we can have is a straight line. Look at the graph below - it represents an object with a starting velocity of v$_{i}$, accelerating to a final velocity v$_{f}$ over a total time t.

To calculate the final displacement we must calculate the area under the graph - this is just the area of the rectangle added to the area of the triangle. This portion of the graph has been shaded for clarity.

Area △ = \[\dfrac{1}{2}\]b.h = \[\dfrac{1}{2}\]t.(v$_{f}$ - v$_{i}$)
Area □ = l.b = t.v$_{i}$
Displacement = Area □ + Area △ => Δx = \[\dfrac{v_i + v_f}{2}\]t​

Derivation of Equation 3.3
This equation is simply derived by eliminating the final velocity vf in equation 3.2. Remembering from equation 3.1 that

v$_{f}$ = v$_{i}$ + at​

then equation 3.2 becomes

Δx = v$_{i}$t + \[\dfrac{1}{2}\]at²​

Derivation of Equation 3.4
This equation is just derived by eliminating the time variable in the above equation. From Equation 3.1 we know

t = \[\dfrac{v_f - v_i}{a}\]
=> 2a.Δx = v$_{f}$² - v$_{i}$² => v$_{f}$² = v$_{i}$² + 2a. Δx​

This gives us the final velocity in terms of the initial velocity, acceleration and displacement and is independent of the time variable.
Exercise: Acceleration
E - 1. A car starts off at 10 m·s$^{−1}$ and accelerates at 1 m·s$^{−2}$ for 10 s. What is its final velocity?
E - 2. A train starts from rest, and accelerates at 1 m·s$^{−2}$ for 10 s. How far does it move?
E - 3. A bus is going 30 m·s$^{−1}$ and stops in 5 s. What is its stopping distance for this speed?
E - 4. A racing car going at 20 m·s$^{−1}$ stops in a distance of 20 m. What is its acceleration?
E - 5. A ball has a uniform acceleration of 4 m·s$^{−1}$. Assume the ball starts from rest. Determine the velocity and displacement at the end of 10 s.
E - 6. A motorcycle has a uniform acceleration of 4 m·s$^{−1}$. Assume the motorcycle has an initial velocity of 20 m·s$^{−1}$. Determine the velocity and displacement at the end of 12 s.
E - 7. An aeroplane accelerates uniformly such that it goes from rest to 144 km·h$^{−1}$in 8s. Calculate the acceleration required and the total distance that it has traveled in this time.

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Physics 10.II Motion in One Dimension

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