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304-F: Kinematic Equations 5: The first kinematic equation in two directions

• Topic Cluster: Kinematics
• Topic: Forward and Backwards Kinematics Quantitative
• Objective: Solve the first kinematic equation in two-directions; use relevant quantities to identify what direction an object is moving and whether is is speeding up or slowing down.
• Content:
• Level: 2

BACK to Ladder Forward and Backwards Kinematics Quantitative

In the unit on forward quantitative kinematics, you learned to use the four kinematic equations to describe constant acceleration motion in one direciton. Now, we will begin learning how to use these equations to reflect motion in two directions: forward and backwards. Mathematically, this is not that difficult: it invovles the same equations, now with both positive and negative numbers. However, interpreting what each positive and negative sign means, and connecting the mathematical result to an understandable, physical result can be quite difficult.

To begin the process of using and interpreting the kinematic equations with positive and negative signs, we will begin with the simplest kinematic equation: the definition of acceleration.

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

Symbol
Quantity
SI Unit
Vector or Scalar
$$v_f$$
Final Velocity
m/s
vector
$$v_i$$
Initial Velocity
m/s
vector
$$a$$
Acceleration
m/s/s
vector
$$\Delta t$$
Time Interval
seconds (s)
scalar
• A vector is a quantity that includes both magnitude and direction.
• In the equation above, initial velocity, final velocity, and acceleration are vectors.
• In one-dimensional problems (in this pod), the direction is represented by a positive or negative sign, and the magnitude is represented by the absolute value of a quantity.

The magnitude of the velocity vector is called the speed. It can be found by taking the absolute value of the velocity.

Problems

Steps to These Problems
1. Whenever solving a problem, pick one direction to be the POSITIVE direction. The opposite direction is the NEGATIVE direction. (Conventionally, right is positive and left is negative, but you are free to reverse this convention if you have any reason to. Always explicitly indicate which direction you chose to be positive.
3. Determine the final velocity using the formula: $$v_f = v_i + a \cdot \Delta t$$
4. Take the aboslute value of the final velocity to determine the magnitude of velocity (called the speed).
6. Determine if the object is speeding up or slowing down:
• If the final velocity and acceleration are in the same direction, then the object is speeding up.
• If the final velocity and acceleration are in the opposite directions, then the object is slowing down.
Instananeous Stops

COMING SOON!

When an object has a velocity of zero for only a moment, it is instaneously stantionary. Typically, objects are instantaneously stationary when they are turning around.

Drawing Quantitative Velocity Graphs

Should this be moved to a different pod?

(this equation needs to be paired with quantiative velocity graphs)

The equation $$v_f = v_i + a \cdot \Delta t$$ can be rewritten in its functional form: $$v(t) = v_0 + a t$$