Easy to perform quantitative experiments at home What are some easy to perform physics experiments that can be done at home (with not too much special equipment) and that allow to actually measure/plot data and draw conclusions from it?
My son is pretty interested in everything physics and we've done just about all possible (qualitative) experiments. He has an idea about some basic physical quantities and I would like to introduce him to the joy of measuring data, plotting them and connecting the measured data to a (preferably simple) formula.
 A: Electricity has a good potential for attractive experiments, and as long as you stay with direct current the math is pretty simple. Plus, measuring I–V (current vs. voltage) characteristics requires only a simple multimeter.
For example, you could show how to tell apart LEDs of different colors by measuring their threshold voltage, or calculate the temperature by measuring diode voltage drop.
A: Try some optics.


*

*You could introduce your son to Snell's Law and the concept of the refractive index. Using a laser pointer and a slab of glass (or an aquarium or vase) you can trace an incoming light ray on a piece of paper and measure the angle of refraction with a protractor. From this you can explain optical density and the sin(x) function (and its relation to a circle). Try to get a total internal reflection going and try it out with different materials (plastic, water, glass, etc) or light sources (green and red laser pointer). A great setup is shown in this YouTube video.


*Explore the lens equation. Find a suitable (convex) lens, a light bulb (with filament, mercury tube or some LED array) and a flat white screen (paper taped to a book cover, doesn't bend) and try to find a position where the screen shows a clear image of the filament of the light bulb. Measure the distances between the light source and the lens ($o$) and between the lens and the image ($i$). Set these lengths into a relation (e.g. by plotting them) and determine the focal length $f$ of the lens from these. The relationship is given by $$\frac{1}{o} + \frac{1}{i} = \frac{1}{f}$$ Experiment with different distances, objects and lenses. Learn more here: Image Formation by Lenses and the Eye (Hyperphysics)


*Explain basic wave optics by showing interference of a laser on a CD. Simply shine with a laser pointer at a CD, so that the reflected light hits a screen (e.g. wall). Notice the symmetry of the maxima; measure the distance between them. Find the distance between the grooves in the CD from Bragg diffraction. (This might actually be a difficult piece of math and physics at such an age, but nevertheless quite fascinating and not as obvious as other optical experiments).


*Steal two pairs of 3D glasses from the movie theatre and discover polarisation. Use your phone's brightness sensor, a toilet paper* tube and the polarising film in the 3D glasses to measure the intensity transmitted through one layer of the filter. Add a second layer and measure the transmission depending on the angle between the two films. Plot it. Again it might be useful to plot the intensity against the sine of the angle. From there you are able to explain the wave nature of light, how it has electric and magnetic components at right angles to each other. You can look at different light sources to discover polarised sources (LCD screens etc). Combining it with the first experiment you could try to find the Brewster angle. (*the toilet paper roll keeps background light away from the sensor)

A: You can build a coil gun with a battery, two switches, a capacitor and an inductor: a bunch of wire wrapped around a straw. One switch charges the cap, one switch discharges the cap through the inductor, and the ball bearing / nail / whatever you put inside the straw goes flying.
Schematic here:
https://ericlippert.com/2013/04/09/tabletop-coilgun/
This lends itself to a lot of possible quantitative experiments regarding the distance the projectile is thrown:


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*Does the number of wraps matter? 

*Does the direction of wraps matter?  

*Does the length of the tube matter?

*What if you have multiple batteries; should they be wired in sequence or parallel?  

*What if you have multiple caps? sequence or parallel?

*What if the tube's angle of elevation is changed? What angle maximizes distance thrown?


Remember that charged capacitors can be dangerous; learn how to discharge them safely. And wear safety goggles when launching ball bearings around the room!
A: You should be able to do quite a bit on the heat capacity of water using just an electric kettle and a cheap digital thermometer. (A cheap plastic jug kettle is probably best, and a 0-100 thermometer not a clinical one). 
Half-fill a kettle with measured amount of fridge-cold water, measure temperature at start, turn on electricity for maybe ten seconds (timed), turn off, swirl to mix water, wait a bit, swirl again, measure temperature. Repeat several times until water too hot to handle safely. Plot graph. 
To a fairly good approximation, energy in and temperature rise are proportional. It is much more challenging to determine to what extent this is not true, and whether this is due to heat leaking in or out as the water gets hotter than room temperature, or due to the water itself changing as it gets hotter. Both are true.
Study of experimental errors. Repeat with more cold water. How repeatable are your observations. What can make them better?  How accurately can a human being with a watch time ten seconds of "on"? (I once read in a book, 0.2s - BS! I'm a musician. Getting fingers round ten notes per second is a major challenge but it's easy to hear if something is a tenth of a second off the beat -- so you should be able to know if the switch clicks are early or late compared to the watch ticks, and target 0.05s sigma? Digression, or another home experiment?)
So, you just might might be able to spot the changes in the heat capacity of water at different temperatures. IIRC about 0.75% change between 5C and 40C, most marked at the cold end. If you can start with water at 1C it's a somewhat bigger percentage change. Not easy to spot with such simple apparatus, but maybe just possible.  
Obligatory safety note: supervise your child (hot water is dangerous, but explain the danger, and let him swirl cold or luke-warm kettles which can't do any more harm than wet clothes and floor. Knowing how to do experiments safely is important. I once came closer to killing myself than I like to recall, now the lesson has sunk in.)
And stop if your child is getting bored.
BTW it's not hard to build a continuous-flow liquid calorimiter at home and nail down the varying heat capacity of water, but that'd probably a bit too complicated for now.
A: I recommend the example I wrote about here before, regarding chocolate bars and microwaves,  not exactly what you are asking for but pure physics all the same: If it does not go as planned, you can eat the test sample, which is a rarity in experiments.

Chocolate And the Speed of Light 
The second (datapoint) example is to measure the diminution of sound when the air is taken out of a container, (bell jar type idea), how easy this is to set up I don't know. Boiling water into steam inside a strong walled container will produce a tighter and tighter vacuum, 
YouTube has lots of example, but they are usually more drama than data.
A: The simple pendulum experiment is very simple to perform, from which a lot of conclusions can be drawn. An object like a ball, like an apple can be used as a weight at the bottom, and a string can be attached to the bob. You could try changing the masses, length of the string and then the amplitude of the oscillations. A lot can be learned from this experiment, and data can be gathered and plotted
A: I) Acceleration due to gravity on objects of different weight
Aim:
To find a relation between the weight of the object and the time taken for the object to reach the ground from a common height. You can optionally plot a graph. Optionally the goal can also be to prove the 60 years old Aristotle wrong.
Apparatus Required:


*

*Stopwatch (clock or timer)

*Test objects


Procedure:


*

*Find a suitable place to drop objects from (common height).

*With stopwatch in one hand and the object in the other hand, drop the object from the common height and start the stop watch simultaneously.

*Stop the clock after the object hits the ground.

*Repeat the experiment three times for each object (explain your son the importance of taking average)

*Note down the values in a table.


Expected Result
The time taken by objects to fall from the same height is independent of its weight.
Notes:


*

*Make use of heavy objects (stones, rocks; must be heavy enough for air resistance to be negligible)

*Ensure that the common height from where the objects are dropped is at least $20m$. For a height of $20m$, the object should theoretically take 2 seconds to fall. For a height of $5m$, the object takes just one second. The larger the height, the better.

*You can use an inclined plane to demonstrate the same concept but frictional forces might start to cause trouble.



II) Relationship between weight/force and extension of a spring
Aim:
To find the relationship between the extension of a spring to the force (weight) applied to the string. Optionally, find the spring constant of the spring.
Apparatus Required:


*

*Spring (which obeys Hooke's law)

*Sample Weights (use multiples of a number; for example, $50g$, $100g$, $150g$, $200g$, $250g$)

*Pan or a weight hanger to hold the weights

*Long measurement scale


Diagram:

Formula:
$$F = -k\Delta x$$ where $F$ is the force applied on the spring, $k$ is the spring constant and $\Delta x$ is the extension.
Procedure:


*

*Suspend the spring from a support

*Attach the hanger/pan to the spring

*Measure the default length of the spring ($l_0$)

*Add weight $w_1$ onto the hanger/pan

*Measure the new length of the spring ($l_1$)

*Add weight $w_2$ onto the hanger/pan

*Measure the new length of the spring ($l_2$)

*Repeat the previous two steps for other weights

*Record the data in a table

*For each value of $l_i$, calculate the extension $\Delta L_1 = l_1 - l_0$

*Plot a graph of the extension (y-axis) vs weight (x-axis)

*Calculate the slop of the graph (the slope of the graph is equal to $\frac{1}{k}$)


Expected Result:
You obtain a straight line in the graph which if extrapolated passes through the origin. This shows that the extension in the spring varies linearly with the force (weight) applied on the spring.
Notes:


*

*Do not use very heavy weights. Other than giving wrong readings, the spring could be permanently damaged.

A: There is an interesting and simple experiment which is to measure the length of one's shadow at different times of the day, and check to see is the shortest shadow does really occur about the nominal noon. It's a fun thing to do by tracing with coloured chalk the outline of the shadow on a driveway or in the street (if possible).  Adults can do this with their kids, comparing shadow lengths.
Depending on how sophisticated you want to get, this can be combined with measurement of the length of the shadow of other objects, like trees or a house or a nearby building.  Using similar triangle "a la Musgrave Ritual"
 
and the known height of your son, one can determine using ratios the height of the building or the tree.
A: Computing $\pi$:
Simply measure the diameter and the circumference of several circles you find in the house (glasses, bowls, tires, etc.). Compute the ratios of the circumferences and the diameters of the circles and they should be hovering around the same value of $$\pi = 3.141..$$
A: Measuring terminal velocity
Get something light like the paper cup-cake cases and drop them from a height.
Film their fall with a tape-measure behind them.
You can add weight (perhaps varying lumps of blue-tac) and watch the terminal velocity change.
You'll want a fairly long distance for the cup-cake cases to fall if you're adding any considerable weight to them but even slight changes should give you different terminal velocities.
This way you can watch the footage back and mark the time when they pass each 10cm marker. This introduces the speed=distance (10cm)/time(however long you measured) and then if you compare each different speed you can plot an acceleration. (Decreasing the distance you compare - the 10cm - will give you more data and a more precise measure and also give you son a little bit of an introduction to the idea of calculus $v=\frac{dx}{dt}$).
Hopefully this has enough in it for your son to find his level.
