1 / 5

Kinematics — Motion in One Dimension

24 min Physics Math

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)

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

Example 2

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

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

Example 3

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

At max height $v = 0$: $0 = 20 - 10t$ → $t = 2$ s.

Max height: $h = 20(2) - \tfrac{1}{2}(10)(4) = 40 - 20 = 20$ m.

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