Training Physics Math Kinematics — Motion in One Dimension
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Kinematics — Motion in One Dimension

24 min Physics Math

Kinematics is the branch of classical mechanics that describes the motion of objects without considering the forces that cause them. By defining quantities like displacement, velocity, and acceleration, we can predict where an object will be at any future time. The core kinematic equations — built from constant-acceleration assumptions — form the backbone of every introductory physics course and appear throughout engineering, sports science, and automotive safety analysis. Mastering these relationships lets you solve problems ranging from a car braking to a stop to a spacecraft adjusting its orbit.

Kinematics

The study of motion uses algebra and quadratic equations to predict where, when, and how fast objects move.

Kinematic Equations (constant acceleration)

$$v = v_0 + at$$

$$x = x_0 + v_0 t + \tfrac{1}{2}at^2$$

$$v^2 = v_0^2 + 2a(x - x_0)$$

Example 1

A ball is dropped from 80 m. How long to hit the ground? ($g = 10$ m/s²)

  1. $80 = \tfrac{1}{2}(10)t^2$ → $t^2 = 16$ → $t = 4$ s.
Example 2

A car accelerates from rest at 3 m/s² for 8 s. How far does it travel?

  1. $x = 0 + 0 + \tfrac{1}{2}(3)(64) = 96$ m.
Example 3

A ball is thrown upward at 20 m/s. When does it reach maximum height? ($g = 10$ m/s²)

  1. At max height $v = 0$:
  2. $0 = 20 - 10t$ → $t = 2$ s.
  3. Max height: $h = 20(2) - \tfrac{1}{2}(10)(4) = 40 - 20 = 20$ m.

Practice Problems

1. A stone falls for 3 s. How far? ($g = 10$ m/s²)
2. A car going 30 m/s brakes at $-5$ m/s². How far until it stops?
3. An object is thrown up at 25 m/s. Find max height.
4. A car accelerates from 10 m/s to 30 m/s in 5 s. Find acceleration.
5. How long does an object take to fall 125 m? ($g = 10$ m/s²)
6. A ball is thrown down at 5 m/s from 45 m. When does it hit ground?
Show Answer Key

1. $\tfrac{1}{2}(10)(9) = 45$ m

2. $v^2 = v_0^2 + 2ax$ → $0 = 900 - 10x$ → $x = 90$ m

3. $v^2 = v_0^2 - 2gh$ → $h = 625/20 = 31.25$ m

4. $a = (30-10)/5 = 4$ m/s²

5. $125 = 5t^2$ → $t = 5$ s

6. $45 = 5t + 5t^2$ → $t^2 + t - 9 = 0$ → $t \approx 2.54$ s

🚀 Kinematic Equations Explorer
Final velocity (v = v₀ + at)
Displacement (d = v₀t + ½at²)
v² = v₀² + 2ad check
Newton's Laws and Forces