# What is momentary acceleration

## Content: Integration: distance, speed, acceleration

Calculation tip:

$$ v (t) = \ int a (t) \ cdot dt $$ $$ v (t) = \ int 5.5 \ cdot dt $$ $$ v (t) = 5.5 \ cdot t + v_0 $$ $$ v (t) = 5.5 \ cdot t $$

**Answer:** The speed function is $ v (t) = 5.5 \ cdot t $.

b) Use $ v $ to determine the path function $ s $, which indicates the path covered after $ t $ seconds.

Calculation tip:

$$ s (t) = \ int v (t) \ cdot dt $$ $$ s (t) = \ int 5.5t \ cdot dt $$ $$ s (t) = 5.5 \ cdot \ frac {t ^ 2 } {2} + c $$ $$ s (t) = 2.75 \ cdot t ^ 2 $$

**Answer:** The function equation is $ s (t) = 2.75 \ cdot t ^ 2 $.

c) $ 180 $ seconds after the start, the solid fuel boosters are burned out and blown off (source: Wikipedia).

Calculate ...

- ... the height at which the rocket is at this point in time.

Calculation tip:

$$ s (t) = 2.75 \ times t ^ 2 $$ $$ s (180) = 2.75 \ times 180 ^ 2 $$ $$ s (180) = 89 100 m $$

**Answer:** After $ 180 $ seconds, our rocket is at $ 89 100m $.

- ... the speed of the rocket at this point.

Calculation tip:

$$ v (t) = 5.5 \ times t $$ $$ v (180) = 5.5 \ times 180 $$ $$ v (180) = 990 m / s $$

$$ 990 m / s \ rightarrow 3 564 km / h $$

**Answer:** After $ 180 $ seconds, our rocket reached a speed of $ 3,564 km / h $.

d) The rocket reaches geostationary orbit at an altitude of about $ 1000 km $ (source: Wikipedia). Determine the number of seconds it takes to reach that orbit.

Calculation tip:

$$ 1 000 000 = 2.75 \ cdot t ^ 2 $$ $$ t = 603.02 $$

**Answer:** After $ 603.02 $ seconds, our rocket reached $ 1,000,000 meters.

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