Take a picture of a falling object against a ruler! Objects fall at a constant acceleration of g=9.8m/(s^2). The time is calculated as t=sqrt(2d/g). You can see the distance the object has traveled in the exposure. As long as the object hasn't reached terminal velocity, and you're relatively close to sea level, it should produce an accurate enough answer for a photographer's needs. Just make sure to zoom in enough to get an accurate measurement.
Try this in practice. There are two problems: first the equation you give is only correct if the object starts from zero velocity, so you need to time the object release much more precisely than the shutter. Second, the distance goes as t^2, which means it is very sensitive to time differences but only works over a small range of times, for ex in 1 second a dropped object falls 4.9 meters, but in 1/60 second it falls 1.4 mm, which would be hard to measure accurately.
With the phonograph record the angular velocity is constant, which makes the method have a hope of working. Many turntables have a stroboscopic pattern printed around the outside, so you can adjust the speed precisely using the flicker of a lamp at the 50 or 60 Hz AC frequency.