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Kinematics
Kinematics Formulas
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2. A New Way of Solving Problems
So far, we have used graphs and simple formulas to determine characteristics of motion of objects.
There is nothing wrong with this method, but often times we saw a lack of precision, and/or a long list of steps to solving problems
We can solve the same problems using a set of formulas, called “kinematics equations”
We use these formulas with respect to a reference point, and in one dimension

3. Formulas Equation 1: 𝒗 𝒇 = 𝒗 𝒊 +𝐚∆𝒕
Keep in mind that d is really ∆d and t is really ∆t.

4. Formulas What formulas do we have already? v = ∆𝒙 ∆𝒕 a = ∆𝒗 ∆𝒕
We can use these formulas to find missing parameters in our 4 kinematics equations
Remember the steps to solving kinematics problems!
Steps to Solving Kinematics Problems
Draw a picture! (For all physics problems)
Set your reference point
Note the given information
Perform required calculations
Verify your answer

5. Hint
Look for hidden information. The text often gives you clues about one of the variables (usually vi = 0, vf = 0, a = -9.81m/s2).
Words such as “from rest”, “dropped”, “falls”, “complete stop”, and so on, can give you valuable information to solve a problem
Sometimes, you can use more than one of the kinematics formulas to solve the same problem

6. Examples 𝒗 𝒇 𝟐 = 𝒗 𝒊 𝟐 + 𝟐𝐚∆𝒙 𝟎 𝟐 = 𝟖 𝟐 + 𝟐𝐚 𝟑.𝟔𝟒 𝟎 = 𝟔𝟒+𝟕.𝟐𝟖𝐚
1) An astronaut is exploring planet Driedgerius and tries jumping straight up on this low-gravity planet. She is able to quite easily push off from the ground with a velocity of 8.0m/s and jump to a height of 3.64m. What is the acceleration of falling objects on the planet Driedgerius?
Examples
Based off the information given, which kinematics equation should we use?
𝒗 𝒇 𝟐 = 𝒗 𝒊 𝟐 + 𝟐𝐚∆𝒙
𝟎 𝟐 = 𝟖 𝟐 + 𝟐𝐚 𝟑.𝟔𝟒
𝟎 = 𝟔𝟒+𝟕.𝟐𝟖𝐚
- 𝟔𝟒 = 𝟕.𝟐𝟖𝐚
𝐚 = 𝒎 𝒔 𝒔

7. Examples
A speedboat increases its speed uniformly from 20 m/s to 30 m/s in a distance of 200m. Find
(a) the magnitude of its acceleration and
(b) the time it takes the boat to travel the 200-m distance.
Based off the information given, which kinematics equation should we use?
𝒗 𝒇 𝟐 = 𝒗 𝒊 𝟐 + 𝟐𝐚∆𝒙
𝟑𝟎 𝟐 = 𝟐𝟎 𝟐 + 𝟐𝐚(𝟐𝟎𝟎)
900 – 400 = 400𝐚
a = 1.25 𝒎 𝒔 𝟐

8. Example 2, Part B
(b) the time it takes the boat to travel the 200-m distance.
Based off the information given, which kinematics equation should we use?
𝒗 𝒇 = 𝒗 𝒊 +𝐚∆𝒕
30 = ∆𝒕
𝒕 = 8s

9. Example 3 A rock is shot up vertically upward from the edge of the top of the building. The rock reaches its maximum height 2 s after being shot. Them, after barely missing the edge of the building as it falls downward, the rock strikes the ground 8s after it was launched. Find: (a) upward velocity the rock was shot at; (b) the maximum height above the building the rock reaches; and (c) how tall is the building?

10. Example 4
A student throws his test paper up in the air (neglecting air resistance) with an initial speed of 8.10 m/s. How long does it take until the paper is headed downward at a speed of 3.23 m/s?

11. Example 5
A missile is launched into the air at an initial velocity of 80 m/s. It is moving with constant velocity until it reaches 1000m, when the engine fails. 
(a) How long does it take it to reach 1000m?
(b) How high does the missile go?
(c) After the engine fails, how long does it take for it to fall back to the earth? 
(d) How long does it stay in the air?
(e) How fast is it going when it hits the ground?

12. Work Slide for Example 5

13. Example 6
If Michael Jordan has a vertical leap of 1.29 m, then what is his takeoff speed and his hang time (total time to move upwards to the peak and then return to the ground)?

14. Example 7
A bullet is moving at a speed of 367 m/s when it embeds into a lump of moist clay. The bullet penetrates for a distance of m. Determine the acceleration of the bullet while moving into the clay. (Assume a uniform acceleration.)

15. Homework
Assignment: create a multi-stage kinematics situation for homework. It must include at least three different periods that all have different accelerations.
Calculus Challenge Problem
A displacement, s, of an object as a function of time, t, is given by d = 3 𝑡 t
a) Find an expression for the acceleration of the object
b) Explain why this expression indicates that the acceleration is not constant

16. Additional Resources
Come ask us for a textbook for additional reading! It can really help you understand by going deeper into the concepts.
Physics classroom; really great site that gives good techniques and examples for physics concepts and problems.