Study guides › Level 10 · Calculus

Speed and motion

Level 10 · Calculus · for questions like “s(t) = 3t² + 2t meters. How fast is it going at t = 4?”

💡 The idea

• Velocity (speed) is the derivative of distance: v(t) = s′(t).
• At the top of a throw, the velocity is 0.

✏️ Worked examples

1. s(t) = 3t² + 2t at t = 4: v(t) = 6t + 2, so v(4) = 26 m/s.

2. A ball with h(t) = 20t − 5t². When is it highest? h′(t) = 20 − 10t = 0, so t = 2 seconds.

3. How high does it go? h(2) = 40 − 20 = 20 m.

⚠️ Watch out

• Speed at a moment is s′(t), not s(t) ÷ t.

🎯 Try these

1. s(t) = t² + 5t: velocity at t = 3?
2. h(t) = 30t − 5t²: when is it highest?
3. Same ball: how high?

Answers: 1) 11 2) 3 seconds 3) 45 m

This guide is also in the 10MathQ app: tap “📖 Study guide” after a question.