Equations of Motion

velocity

$${ v = \frac{s}{t}}$$

where v is the speed (velocity), s is the distance travelled and t is the time interval.

acceleration

Acceleration is the change of velocity over time. In a time interval, we use the initial velocity and final velocity to approximately calculate the acceleration.

$${a = \frac{v - u}{t}}$$

where a is the acceleration, u is the initial velocity, v is the final velocity and t is the time interval.

From this we get

$${v = u + at}$$

distance travelled with uniform acceleration

$${s = \frac{1}{2}(v + u)t}$$

using the equation for v and substituting

$${s = ut + \frac{1}{2}at^2}$$

Note - in a graph this is the area beneath a trapezium.

velocity, distance and acceleration

$${v^2 = u^2 + 2as}$$

‘initially at rest’ means the initial velocity u = 0.

free fall - a special case

Galileo

Galileo attempted to verify which of the following hypotheses was correct.

${v \sim s}$ or ${v \sim t}$ ?

He actually did not drop something, as time keeping was not accurate enough, but investigated a “slowed down” fall on a very long wooden ramp with a wooden ball rolling down.

In the end, it was

$${v \sim t}$$

and the gravitional constant g is the acceleration. So,

$${v = g \cdot t}$$

and the distance travelled (the object falling is initially at rest) is

$${s = \frac{1}{2}gt^2}$$

dropping string with weights - die Fallschnur

In class it is worth investigating both cases.

Procedure -

  1. drop a string with weights attached at equal distance.
  1. Record the sound the two drops make and compare.

  2. Which string has a regular beat?

Here, are the images recorded with Audacity.

increasing intervals

10, 30, 50, 70 cm

Here, the time intervals are equal. This means the distance travelled in the same time is increasing steadily and therefore the velocity is increasing.

equally spaced intervals

40 cm

Equally spaced distances would have meant the velocity increases proportional to the distance s. But, the time decreases. So, the velocity is increasing in time.

sound file (mp3)

Compare the sound, the first is increasing distances and the second part is when the distances are equal (40 cm).

In the first part the beats are at roughly equal distances. While in the second part the intervals between each beat are shorter and shorter.

conclusion

The velocity v is proportional to time t.

In the first image the distance travelled in the same time is increasing steadily. So, the velocity is increasing steadily. The distance was increased by 20 cm. This implies that the increase in velocity is constant and we have a constant accelation of about $10 \frac{m}{s^2}$.

In reality, in Berlin, we have an acceleration of about $9.81 \frac{m}{s^2}$

horizontal projectile motion - der waagrechte Wurf

see here


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