Motion, speed, and acceleration

Make a guess before you read on. Most people say the bowling ball, because it is heavier. That was also the answer of the Greek philosopher Aristotle, and it was the standard answer in Europe for almost two thousand years.

This lesson answers the question the way physics answers any question: by measuring. Along the way you will learn the five words physics uses to describe motion: position, displacement, speed, velocity, and acceleration. Describing motion with those words is called kinematics (Urone & Hinrichs, 2022).

Physics starts with things you can measure. For motion, that means distance and time. Scientists everywhere measure distance in meters and time in seconds, two of the base units of the International System of Units, called SI units (National Institute of Standards and Technology [NIST], n.d.). One meter is a little more than three feet.

A claim about falling only counts as physics if a measurement could prove it wrong. "Heavier things fall faster" passes that test. You can time two balls and see. "Things fall because it is their nature" does not. No stopwatch reading could count against it.

There are two honest answers, and physics keeps both. Your position is where you are, measured from a starting point you choose. That starting point and the directions you measure from are your reference frame.

The total ground you covered is the distance: 5 blocks plus 2 blocks, or 7 blocks. The change in your position, from start to finish, is the displacement: 3 blocks east. Distance ignores direction. Displacement includes it. Figure 1.1 shows the walk.

Not necessarily. Speed tells how fast something covers ground. Average speed is distance divided by time.

average speed = distance ÷ time

Drive 120 miles in 2 hours and your average speed is 60 miles per hour. Velocity is speed with a direction attached: 60 mph north, or 60 mph south. Two cars at 60 mph heading toward each other have the same speed but opposite velocities (Urone & Hinrichs, 2022).

A number with a direction is called a vector. Displacement and velocity are vectors. A number with no direction, like distance, speed, or temperature, is a scalar. In drawings, physicists show a vector as an arrow. The arrow points in the direction, and its length shows the size. Figure 1.2 uses arrows this way.

Acceleration is how fast velocity changes. It is the change in velocity divided by the time the change takes (Urone & Hinrichs, 2022).

acceleration = change in velocity ÷ time

A car that goes from 0 to 60 mph in 6 seconds gains 10 mph every second. In SI units that is about 4.5 meters per second, every second, written 4.5 m/s² ("meters per second squared").

In everyday talk, "accelerate" means speed up. In physics it means any change in velocity. Speeding up is acceleration. Slowing down is acceleration. And because velocity includes direction, turning at a steady speed is acceleration too. That is the car on the right in Figure 1.2.

It speeds up at a steady rate. Near Earth's surface, a dropped object with little air resistance gains about 9.8 meters per second of speed every second. This value is written g, the acceleration due to gravity, and it is about 9.8 m/s² (Urone & Hinrichs, 2022). Motion where gravity is the only thing acting is called free fall.

After 1 second, a dropped ball moves at 9.8 m/s, about 22 mph. After 2 seconds, 19.6 m/s. After 3 seconds, 29.4 m/s. The speed grows in equal steps.

The distance does not grow in equal steps. It grows with the square of the time. The ball falls about 4.9 meters in the first second, 19.6 meters by the end of 2 seconds, and 44.1 meters by the end of 3 seconds.

distance fallen = ½ × g × time²

Figure 1.3 shows this as a strobe picture: the ball's position every tenth of a second. Notice that the gaps keep getting bigger.

A famous story says Galileo Galilei dropped two balls from the Leaning Tower of Pisa. That story comes from a biography written by his student, and historians are not sure it happened as told. What Galileo did write about, in his 1638 book Two New Sciences, is something cleverer.

Falling is too fast to time by eye. So Galileo "slowed gravity down" by rolling bronze balls down a long, smooth, gently sloped ramp. He timed the rolls with a water clock: he let water flow out of a large container during each roll and weighed the water that came out (Galilei, 1638/1914).

He found that the distance covered grew with the square of the time. In equal time steps, the ball covered distances in the ratio 1, 3, 5, 7, the odd numbers. Add them up and you get 1, 4, 9, 16, the squares. Figure 1.5 lays out that pattern. It held for every slope he tried, and a steeper ramp only made it faster. His conclusion was that free fall is the same pattern, sped up.

If gravity were the only thing acting, neither. Every object would fall with the same acceleration, g, no matter its weight. The heavier ball gets a stronger pull from gravity, but it also takes more force to get moving, in exactly the same proportion. Lesson 2 shows why those two effects cancel.

On Earth there is also air resistance, the push of air against a moving object. For dense objects like these two balls, it is small over a short drop. Run the numbers for a drop from 20 meters (about a six-story building) and the bowling ball lands in about 2.02 seconds. The baseball lands a few hundredths of a second later, less than a meter behind. For a feather, air resistance wins, and the drop takes more than half a minute.

Try it yourself in Figure 1.4. Drop all three with the air on, then switch the air off and drop them again.

Yes, on the Moon, which has almost no atmosphere. In 1971, at the end of the Apollo 15 mission, astronaut David Scott held out a 1.32-kilogram aluminum geology hammer and a falcon feather and let go of both at once, on live television. They hit the ground together (Williams, 2023).

They fell more slowly than they would on Earth, because the Moon's gravity is weaker: about 1.62 m/s², roughly one sixth of Earth's (Williams, n.d.). But they fell together. Galileo's conclusion held up, 333 years later, on another world.

This lesson described motion. It did not explain it. Why does a ball speed up as it falls? Why does a moving puck slide on and on across the ice? Lesson 2 brings in forces and Newton's three laws of motion.

And here is something to look forward to. The same g that pulls a ball to the floor, weakened by distance, keeps the Moon in its orbit. In Lesson 4 you will see Isaac Newton work that out, and it is one of the most remarkable discoveries anyone has made about the universe.