guide

Motion With Constant Acceleration (Examples, Exercises, Solutions)

An abstraction often used in physics is motion with constant acceleration. This is a good approximation for many different situations: free fall over small distances or in low-density atmospheres, full braking in car traffic, an object sliding down an inclined plane, etc … The mathematics behind this special case is relatively simple. Assume the object that is subject to the constant acceleration a (in m/s²) initially has a velocity v(0) (in m/s). Since the velocity is the integral of the acceleration function, the object’s velocity after time t (in s) is simply:

1) v(t) = v(0) + a · t

For example, if a car initially goes v(0) = 20 m/s and brakes with a constant a = -10 m/s², which is a realistic value for asphalt, its velocity after a time t is:

v(t) = 20 – 10 · t

After t = 1 second, the car’s speed has decreased to v(1) = 20 – 10 · 1 = 10 m/s and after t = 2 seconds the car has come to a halt: v(2) = 20 – 10 · 2 = 0 m/s. As you can see, it’s all pretty straight-forward. Note that the negative acceleration (also called deceleration) has led the velocity to decrease over time. In a similar manner, a positive acceleration will cause the speed to go up. You can read more on acceleration in this blog post.

What about the distance x (in m) the object covers? We have to integrate the velocity function to find the appropriate formula. The covered distance after time t is:

2) x(t) = v(0) · t + 0.5 · a · t²

While that looks a lot more complicated, it is really just as straight-forward. Let’s go back to the car that initially has a speed of v(0) = 20 m/s and brakes with a constant a = -10 m/s². In this case the above formula becomes:

x(t) = 20 · t – 0.5 · 10 · t²

After t = 1 second, the car has traveled x(1) = 20 · 1 – 0.5 · 10 · 1² = 15 meters. By the time it comes to a halt at t = 2 seconds, it moved x(2) = 20 · 2 – 0.5 · 10 · 2² = 20 meters. Note that we don’t have to use the time as a variable. There’s a way to eliminate it. We could solve equation 1) for t and insert the resulting expression into equation 2). This leads to a formula connecting the velocity v and distance x.

3) Constant acceleration_html_b85f3ec

Solved for x it looks like this:

3)’ Constant acceleration_html_m23bb2bb3

It’s a very useful formula that you should keep in mind. Suppose a tram accelerates at a constant a = 1.3 m/s², which is also a realistic value, from rest (v(0) = 0 m/s). What distance does it need to go to full speed v = 10 m/s? Using equation 3)’ we can easily calculate this:

Constant acceleration_html_m11de6604

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Here are a few exercises and solutions using the equations 1), 2) and 3).

1. During free fall (air resistance neglected) an object accelerates with about a = 10 m/s. Suppose the object is dropped, that is, it is initially at rest (v(0) = 0 m/s).

a) What is its speed after t = 3 seconds?
b) What distance has it traveled after t = 3 seconds?
c) Suppose we drop the object from a tower that is x = 20 meters tall. At what speed will it impact the ground?
d) How long does the drop take?

Hint: in exercise d) solve equation 1) for t and insert the result from c)

2. During the reentry of space crafts accelerations can be as high as a = -70 m/s². Suppose the space craft initially moves with v(0) = 6000 m/s.

a) What’s the speed and covered distance after t = 10 seconds?
b) How long will it take the space craft to half its initial velocity?
c) What distance will it travel during this time?

3. An investigator arrives at the scene of a car crash. From the skid marks he deduces that it took the car a distance x = 55 meters to come to a halt. Assume full braking (a = -10 m/s²). Was the car initially above the speed limit of 30 m/s?

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Solutions to the exercises:

Exercise 1

a) 30 m/s
b) 45 m
c) 20 m/s
d) 2 s

Exercise 2

a) 5,300 m/s and 56,500 m
b) 42.9 s (rounded)
c) 192,860 m (rounded)

Exercise 3

Yes (he was initially going 33.2 m/s)

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To learn the basic math you need to succeed in physics, check out the e-book “Algebra – The Very Basics”. For an informal introduction to physics, check out the e-book “Physics! In Quantities and Examples”. Both are available at low prices and exclusively for Kindle.

Audio Effects: All About Compressors

Almost all music and recorded speech that you hear has been sent through at least one compressor at some point during the production process. If you are serious about music production, you need to get familiar with this powerful tool. This means understanding the big picture as well as getting to know each of the parameters (Threshold, Ratio, Attack, Release, Make-Up Gain) intimately.

  • How They Work

Throughout any song the volume level varies over time. It might hover around – 6 dB in the verse, rise to – 2 dB in the first chorus, drop to – 8 dB in the interlude, and so on. A term that is worth knowing in this context is the dynamic range. It refers to the difference in volume level from the softest to the loudest part. Some genres of music, such as orchestral music, generally have a large dynamic range, while for mainstream pop and rock a much smaller dynamic range is desired. A symphony might range from – 20 dB in the soft oboe solo to – 2 dB for the exciting final chord (dynamic range: 18 dB), whereas your common pop song will rather go from – 8 dB in the first verse to 0 dB in the last chorus (dynamic range: 8 dB).

During a recording we have some control over what dynamic range we will end up with. We can tell the musicians to take it easy in the verse and really go for it in the chorus. But of course this is not very accurate and we’d like to have full control of the dynamic range rather than just some. We’d also like to be able to to change the dynamic range later on. Compressors make this (and much more) possible.

The compressor constantly monitors the volume level. As long as the level is below a certain threshold, the compressor will not do anything. Only when the level exceeds the threshold does it become active and dampen the excess volume by a certain ratio. In short: everything below the threshold stays as it is, everything above the threshold gets compressed. Keep this in mind.

Suppose for example we set the threshold to – 10 dB and the ratio to 4:1. Before applying the compressor, our song varies from a minimum value of – 12 dB in the verse to a maximum value – 2 dB in the chorus. Let’s look at the verse first. Here the volume does not exceed the threshold and thus the compressor does not spring into action. The signal will pass through unchanged. The story is different for the chorus. Its volume level is 8 dB above the threshold. The compressor takes this excess volume and dampens it according to the ratio we set. To be more specific: the compressor turns the 8 dB excess volume into a mere 8 dB / 4 = 2 dB. So the compressed song ranges from – 12 dB in the verse to -10 dB + 2 dB = – 8 dB in the chorus.

Here’s a summary of the process:

Settings:

Threshold: – 10 dB
Ratio: 4:1

Before:

Minimum: – 12 dB
Maximum: – 2 dB
Dynamic range: 10 dB

Excess volume (threshold to maximum): 8 dB
With ratio applied: 8 dB / 4 = 2 dB

After:

Minimum: – 12 dB
Maximum: – 8 dB
Dynamic range: 4 dB

As you can see, the compressor had a significant effect on the dynamic range. Choosing appropriate values for the threshold and ratio, we are free to compress the song to any dynamic range we desire. When using a DAW (Digital Audio Workstation such as Cubase, FL Studio or Ableton Live), it is possible to see the workings of a compressor with your own eyes. The image below shows the uncompressed file (top) and the compressed file (bottom) with the threshold set to – 12 dB and the ratio to 2:1.

MIXING_html_26d7be80

The soft parts are identical, while the louder parts (including the short and possibly problematic peaks) have been reduced in volume. The dynamic range clearly shrunk in the process. Note that after applying the compressor, the song’s effective volume (RMS) is much lower. Since this is usually not desired, most compressors have a parameter called make-up gain. Here you can specify by how much you’d like the compressor to raise the volume of the song after the compression process is finished. This increase in volume is applied to all parts of the song, soft or loud, so there will not be another change in the dynamic range. It only makes up for the loss in loudness (hence the name).

  • Usage of Compressors

We already got to know one application of the compressor: controlling the dynamic range of a song. But usually this is just a first step in reaching another goal: increasing the effective volume of the song. Suppose you have a song with a dynamic range of 10 dB and you want to make it as loud as possible. So you move the volume fader until the maximum level is at 0 dB. According to the dynamic range, the minimum level will now be at – 10 dB. The effective volume will obviously be somewhere in-between the two values. For the sake of simplicity, we’ll assume it to be right in the middle, at – 5 dB. But this is too soft for your taste. What to do?

You insert a compressor with a threshold of – 6 dB and a ratio of 3:1. The 4 dB range from the minimum level – 10 dB to the threshold – 6 dB is unchanged, while the 6 dB range from the threshold – 6 dB to the maximum level 0 dB is compressed to 6 dB / 3 = 2 dB. So overall the dynamic range is reduced to 4 dB + 2 dB = 6 dB. Again you move the volume fader until the maximum volume level coincides with 0 dB. However, this time the minimum volume will be higher, at – 6 dB, and the effective volume at – 3 dB (up from the – 5 dB we started with). Mission accomplished, the combination of compression and gain indeed left us with a higher average volume.

In theory, this means we can get the effective volume up to almost any value we desire by compressing a song and then making it louder. We could have the whole song close to 0 dB. This possibility has led to a “loudness war” in music production. Why not go along with that? For one, you always want to put as much emphasis as possible on the hook. This is hard to do if the intro and verse is already blaring at maximum volume. Another reason is that severely reducing the dynamic range kills the expressive elements in your song. It is not a coincidence that music which strongly relies on expressive elements (orchestral and acoustic music) usually has the highest dynamic range. It needs the wide range to go from expressing peaceful serenity to expressing destructive desperation. Read the following out loud and memorize it: the more expression it has, the less you should compress. While a techno song might work at maximum volume, a ballad sure won’t.

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Background Info – SPL and Loudness

Talking about how loud something is can be surprisingly complicated. The problem is that our brain does not process sound inputs in a linear fashion. A sound wave with twice the sound pressure does not necessarily seem twice as loud to us. So when expressing how loud something is, we can either do this by using well-defined physical quantities such as the sound pressure level (which unfortunately does not reflect how loud a person perceives something to be) or by using subjective psycho-acoustic quantities such as loudness (which is hard to define and measure properly).

Sound waves are pressure and density fluctuations that propagate at a material- and temperature-dependent speed in a medium. For air at 20 °C this speed is roughly 340 m/s. The quantity sound pressure expresses the deviation of the sound wave pressure from the pressure of the surrounding air. The sound pressure level, in short: SPL, is proportional to the logarithm of the effective sound pressure. Long story short: the stronger the sound pressure, the higher the SPL. The SPL is used to objectively measure how loud something is. Another important objective quantity for this purpose is the volume. It is a measure of how much energy is contained in an audio signal and thus closely related to the SPL.

A subjective quantity that reflects how loud we perceive something to be is loudness. Due to our highly non-linear brains, the loudness of an audio signal is not simply proportional to its SPL or volume level. Rather, loudness depends in a complex way on the SPL, frequency, duration of the sound, its bandwidth, etc … In the image below you can see an approximation of the relationship between loudness, SPL and frequency.

MIXING_html_mc95d258

Any red curve is a curve of equal loudness. Here’s how we can read the chart. Take a look at the red curve at the very bottom. It starts at 75 dB SPL and a frequency of 20 Hz and reaches 25 dB SPL at 100 Hz. Since the red curve is a curve of equal loudness, we can conclude we perceive a 75 dB SPL sound at 20 Hz to be just as loud as a 25 dB SPL sound at 100 Hz, even though from a purely physical point of view the first sound is three times as loud as the second (75 dB / 25 dB = 3).

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MIXING_html_m53a1053e

(Compressor in Cubase)

  • Threshold and Ratio

What’s the ideal threshold to use? This depends on what you are trying to accomplish. Suppose you set the threshold at a relatively high value (for example – 10 dB in a good mix). In this case the compressor will be inactive for most of the song and only kick in during the hook and short peaks. With the threshold set to a high value, you are thus “taking the top off”. This would be a suitable choice if you are happy with the dynamics in general, but would like to make the mix less aggressive.

What about low thresholds (such as -25 dB in a good mix)? In this case the compressor will be active for the most part of the song and will make the entire song quite dense. This is something to consider if you aim to really push the loudness of the song. Once the mix is dense, you can go for a high effective volume. But a low threshold compression can also add warmth to a ballad, so it’s not necessarily a tool restricted to usage in the loudness war.

Onto the ratio. If you set the ratio to a high value (such as 5:1 and higher), you are basically telling the mix: to the threshold and no further. Anything past the threshold will be heavily compressed, which is great if you have pushy peaks that make a mix overly aggressive. This could be the result of a snare that’s way too loud or an inexperienced singer. Whatever the cause, a carefully chosen threshold and a high ratio should take care of it in a satisfying manner. Note though that in this case the compressor should be applied to the track that is causing the problem and not the entire mix.

A low value for the ratio (such as 2:1 or smaller) will have a rather subtle effect. Such values are perfect if you want to apply the compressor to a mix that already sounds well and just needs a finishing touch. The mix will become a little more dense, but its character will be kept intact.

  • Attack and Release

There are two important parameters we have ignored so far: the attack and release. The attack parameter allows you to specify how quickly the compressor sets in once the volume level goes past the threshold. A compressor with a long attack (20 milliseconds or more) will let short peaks pass. As long as these peaks are not over-the-top, this is not necessarily a bad thing. The presence of short peaks, also called transients, is important for a song’s liveliness and natural sound. A long attack makes sure that these qualities are preserved and that the workings of the compressor are less noticeable.

A short attack (5 milliseconds or less) can produce a beautifully crisp sound that is suitable for energetic music. But it is important to note that if the attack is too short, the compressor will kill the transients and the whole mix will sound flat and bland. Even worse, a short attack can lead to clicks and a nervous “pumping effect”. Be sure to watch out for those as you shorten the attack.

The release is the time for the compressor to become inactive once the volume level goes below the threshold. It is usually much longer than the attack, but the overall principles are similar. A long release (600 milliseconds or more) will make sure that the compression happens in a more subtle fashion, while a short release (150 milliseconds or less) can produce a pumping sound.

It is always a good idea to choose the release so that it fits the rhythm of your song (the same of course is true for temporal parameters in reverb and delay). One way to do this is to calculate the time per beat TPB in milliseconds from your song’s tempo as measured in beats per minute BPM and use this value as the point of reference.

TPB [ms] = 60000 / BPM

For example, in a song with the tempo BPM = 120 the duration of one beat is TPB = 60000 / 120 = 500 ms. If you need a longer release, use a multiple of it (1000 ms, 1500 ms, and so on), for a shorter release divide it by any natural number (500 ms / 2 = 250 ms, 500 ms / 3 = 167 ms, and so on). This way the compressor will “breathe” in unison with your music.

If you are not sure where to start regarding attack and release, just make use of the 20/200-rule: Set the attack to 20 ms, the release to 200 ms and work towards the ideal values from there. Alternatively, you can always go through the presets of the compressor to find suitable settings.

 

You can learn about advanced compression techniques as well as other effects from Audio Effects, Mixing and Mastering, available for Kindle for $ 3.95.