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302-E: Kinematic Equations 2: All Four Together

• Topic Cluster: Kinematics
• Topic: Forward Kinematics Quantitative
• Objective: Solve numerical algebraic problems with the kinematic equations; use all four kinematic equations interchangeably; know the acceleration of a falling object.
• Content: The four kinematic equations relate the five kinematic variables for an object moving with a constant acceleration. They can be applied to problems of varying difficulty.
• Level: 2

BACK to Ladder Forward Kinematics Quantitative

In a previous pod, we introduced each the the first three kinematic equations. In this pod, we will use all four kinematic equations to solve problems that are more self-directed.

The kinematic equations are four equations that can be used to relate the different relevant quantities when an object is moving. The four kinematic equations can be used whenever an object is moving with a constant acceleration. They cannot be used otherwise.

The Five Kinematic Quantities:
1. Which kinematic equation does not include the variable $$\Delta t$$?
2. Which kinematic variable has unit of meters per second per second?
3. Which kinetic equation asks you to add final and initial velocity?
4. How many kinematic quantities are in each kinematic equation?

Kinematic Equation 1: The Definition of Acceleration

$$v_f = v_i + a \cdot \Delta t$$

1. I had an initial velocity of 8 m/s and now have a final velocity of 20 m/s. I had an acceleration of 4 m/s2. How much time did I take?
2. I had an initial velocity of 2 m/s and now have a final velocity of 23 m/s. I had an acceleration of 3 m/s2. How much time did I take?

Kinematic Equation 2: The King of Kinematic Equations

Note: "The King of Kinematic Equations" is not the official name, it is a name that I invented. But if you keep learning physics, you will see why this kinematic equation deserves to be crowned the king.

$$\Delta x = v_i \cdot \Delta t + \frac{1}{2}a \left( \Delta t \right)^2$$

1. When I travel with an acceleration of 5 m/s2 for a time of 3 s, I travel a displacement of 69 m. What was my initial velocity?
2. When I travel with an acceleration of 7 m/s2 for a time of 4 s, I travel a displacement of 104 m. What was my initial velocity?
3. I began with an initial velocity of 3 m/s and traveled for a time of 6 s. I eventually traveled a displacement of 126 m. What was my acceleration?
4. I began with an initial velocity of 16 m/s and traveled for a time of 2 s. I eventually traveled a displacement of 50 m. What was my acceleration?

Kinematic Equation 3: The Average Velocity Equation

$$\Delta x = \left( \frac{v_i + v_f}{2} \right) \Delta t$$

1. I begin with an initial velocity of 3 m/s and travel a displacement of 49 m in a time of 7 seconds. What is my final velocity?
2. I begin with an initial velocity of 5 m/s and travel a displacement of 88 m in a time of 8 s. What is my final velocity?
3. I travel a displacement of 90 m in a time of 5 seconds and end with a final velocity of 23 m/s. What was my initial velocity?
4. I travel a displacement of 15 m in a time of 3 s and end with a final velocity of 9 m/s. What was my initial velocity?

Kinematic Equation 4: The No-Time Equation

$$v_f^2 = v_i^2 + 2 a \cdot \Delta x$$

1. I begin with an initial velocity of 3 m/s and accelerate at a rate of 4 m/s2, while traveling a displacement of only 5 meters. What is my final velocity?
2. I begin with an initial velocity of 2 m/s and accelerate at a rate of 2 m/s2 while traveling a displacement of 8 m. What is my final velocity?
3. I travel a displacement of only 4 meters while accelerating at a rate of 7 m/s2. If my final velocity is 9 m/s, what was my initial velocity?
4. I travel a displacement of 6 m while accelerating at a rate of 3 m/s2 and end with a final velocity of 6 m/s. What was my initial velocity?
5. I begin with an initial velocity of 8 m/s and accelerate to a final velocity of 10 m/s. I traveled a displacement of 6 m, what was my acceleration?
6. I begin with an initial velocity of 6 m/s and slow down to a final velocity of 2 m/s. If I travel a displacement of 4 m, what was my acceleration?
[The answer should be negative here!]
7. I begin with an initial velocity of 4 m/s and accelerate to a final velocity of 12 m/s. My acceleration was 2 m/s2. What displacement did I travel?
8. I begin at rest and accelerate at a rate of 4 m/s2. I finish with a final velocity of 20 m/s. What displacement did I travel?