targas-edu.de
DE ☾ A+

Application: the throw

This page was translated from German with AI. It has not been checked by a native speaker. Original page in German

You throw a ball upwards. Its height in metres is \(h(t) = -5t^2 + 10t + 2\). How high does it fly, and when does it land?
📄 Open worksheet (PDF, German)

The video

In the video we read the greatest height at the vertex and calculate the landing with the pq-formula.

Subtitles are available in ten languages. Choose them at the bottom right of the video.

Contents of the video

  1. 0:00 Introduction
  2. 0:22 The greatest height
  3. 0:43 The landing
  4. 1:33 Careful, mistake!
  5. 2:00 Try it yourself

The most important points

Greatest height

\(h(t) = -5(t-1)^2 + 7\), \(S(1\,|\,7)\): after \(1\,\text{s}\) the ball is \(7\,\text{m}\) high. First number: time. Second number: height.

The vertex is the highest point of the flight path.
The vertex is the highest point of the flight path.

Landing

\(h(t) = 0\): \(t^2 - 2t - 0{,}4 = 0\), \(t = 1 \pm \sqrt{1{,}4} \approx 1 \pm 1{,}18\). The negative solution does not fit: the ball lands after about \(2{,}2\,\text{s}\).

The ball lands at t \approx 2{,}18\,\text{s}.
The ball lands at \(t \approx 2{,}18\,\text{s}\).

Careful: \(S(1\,|\,7)\) does not mean “after 7 seconds”. Watch the units: seconds and metres.

The worksheet

Here you can see the worksheet for the video (in German). First it explains everything again, then there are practice tasks at three levels. You can also open it as a PDF and print it.

📄 Open worksheet (PDF, German)

The worksheet – Page 1
Page 1/4
The worksheet – Page 2
Page 2/4
The worksheet – Page 3
Page 3/4
Show the worksheet solutions
The worksheet – Page 4
Page 4/4

Try it yourself

Task

A stone flies with \(h(t) = -5t^2 + 20t\). How high does it fly, and after how many seconds is it back on the ground?

Show solution

\(h(t) = -5(t-2)^2 + 20\): \(20\,\text{m}\) after \(2\,\text{s}\). \(-5t(t-4) = 0\): on the ground after \(4\,\text{s}\).

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: Anwendung quadratischer Funktionen: Wurf (Scheitelpunkt und Nullstelle). Stelle mir nacheinander fünf kurze Aufgaben. Warte immer auf meine Antwort. Sag mir dann, ob sie richtig ist, und erkläre kurz, warum. Benutze einfache deutsche Sätze.

How this was made: The idea, structure and examples of the series are mine. I made them with the help of AI: a language model (LLM) built the slides of the video following my instructions. The voice was generated with Qwen3-TTS, modelled on my own voice.

The subtitles, the translations, the worksheet and this page were also made with the help of AI. The translations have not been checked by native speakers.

The videos, worksheets, texts and prompts in this series are licensed under CC BY 4.0: you may use, change and share them if you name the source.