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Steep and gentle

Change and motion · level 1, lesson 6 of 6

A question to start

At the playground there are two slides. One is short and steep. The other is long and gentle, with lots of gentle twists. Both start from platforms of the same height.

Which slide gets you to the bottom first? And here is a harder one: on which slide are you going faster when you reach the bottom?

The first answer will not surprise you. The second one surprises almost everybody, even grown-ups. Keep it in mind, and we will come back to it at the end of the lesson.

The idea

What "steep" means

A ramp is a flat board or path that goes up at a slant, like a slide, a hill, or a plank leaning on a step.

  • A ramp is steep if it goes up a lot quickly. Walking up it is hard work. Think of a mountain side, or a ladder.
  • A ramp is gentle if it only goes up a little. Walking up it is easy. Think of a slope for a wheelchair or a pram.
  • A flat path does not go up at all.

Steepness is about how much a ramp goes up compared with how much it goes along.

Making a ramp at home

Here is an experiment. You need a long flat board (or a big piece of stiff cardboard, or a tray), some books, and a ball or a toy car.

  1. Put one end of the board on a pile of 2 books. This is a gentle ramp.
  2. Hold the ball at the top and let it go. Do not push it! Watch how fast it rolls.
  3. Now put the same end on a pile of 5 books. This is a steep ramp.
  4. Let the ball go from the top again. Watch.
gentle: 2 books steep: 5 books

What happens? On the steep ramp the ball rolls faster. It gets to the bottom sooner, and it zooms further across the floor afterwards.

Why the steep ramp is faster

Every ball is pulled down towards the ground. That pull is called gravity. It is why things fall when you drop them.

  • If the ramp is very steep, almost straight up and down, the ball is almost falling. Gravity pulls it along the ramp nearly as hard as if you dropped it.
  • If the ramp is gentle, gravity pulls mostly into the board, and only a little bit pulls the ball along the board. So the ball speeds up only slowly.
  • If the board is flat, gravity pulls straight into the board and not along it at all. The ball stays still.

You can feel this with your own body. Stand on a gentle hill and you can stand still easily. On a steep hill you have to lean and hold on, because you feel pulled down the slope. On a flat floor you feel no pull to either side.

So from flat to gentle to steep to straight down, the pull along the ramp gets bigger and bigger, and the ball speeds up more and more quickly.

Measuring steepness with squares

How can we tell which ramp is steeper without trying a ball? We count how far it goes up and how far it goes along.

A: up 2 foreach 1 along B: up 1 foreach 2 along

Ramp A climbs 2 squares up for every 1 square along. Ramp B climbs only 1 square up for every 2 squares along. Ramp A is much steeper.

The fair way to compare is like a fair race (lesson 2): make one thing the same. Walk the same distance along both ramps, say 2 squares. Ramp A climbs 4 squares in that distance; ramp B climbs only 1. More up for the same along means steeper.

Steep going up, steep going down

Steepness matters both ways.

  • Going up a steep hill on a bike, you go slowly, because you are fighting gravity. On a gentle hill you can pedal along more quickly.
  • Going down a steep hill, you go fast, because gravity helps you. Going down a gentle hill, you speed up only a little.

That is why roads on mountains often go back and forth in zigzags. A zigzag road is longer, but it is gentler, so cars and bikes can climb it more easily, and they do not race too fast on the way down.

Rovers on Mars must be careful about steepness too. On a slope that is too steep, a rover could slide on loose sand or even tip over, and there is nobody there to set it upright again.

Back to the slides: the big surprise

Now the question from the start. Two slides start at the same height: one short and steep, one long and gentle.

  1. Who gets to the bottom first? The steep slide. It is shorter, and you speed up more quickly on it.
  2. Who is going faster at the bottom? Here is the surprise: if the slides were perfectly slippery, you would be going the same speed at the bottom of both! On the gentle slide you speed up slowly, but you speed up for a longer time, and it works out the same. What decides your speed at the bottom is how high you started, not how steep the slide was.

Real slides are not perfectly slippery. Your clothes rub on the slide, and a long slide has more rubbing, so in real life the long gentle slide is usually a bit slower at the bottom. But the big idea is true and important: the higher you start, the faster you end up. That is why, in our book experiment, the steep ramp (with its top higher up) sent the ball further across the floor.

Worked examples

Example 1. Which is steeper? Ramp C goes up 3 squares for every 1 square along. Ramp D goes up 1 square for every 1 square along. Which is steeper?

  1. Look at the same "along" for both: 1 square.
  2. C goes up 3; D goes up 1.
  3. C is steeper, because it goes up more for the same along.

Example 2. A toy car race. Two toy cars are let go from the top of two boards of the same length. One board rests on 2 books, the other on 5 books. Which car reaches the floor first?

  1. The board on 5 books is steeper.
  2. On a steeper ramp, gravity pulls more along the board, so the car speeds up more quickly.
  3. The car on the 5-book ramp gets there first.

Example 3. The flat board. You put a ball on a board lying flat on the floor. It does not move. Why?

  1. Gravity pulls the ball straight down, into the board.
  2. On a flat board, none of that pull goes along the board.
  3. So nothing makes the ball start rolling. It stays still.

Example 4. Biking to school. On the way to school, Ana rides down a steep hill. On the way home, she rides up the same hill. When is she faster on that hill?

  1. Going down, gravity helps her: she is fast.
  2. Going up, gravity works against her: she is slow, and it is tiring.
  3. She is faster on the way to school (downhill).

Try it

  1. Which is steeper: a ladder leaning against a wall, or a ramp for a wheelchair?
  2. A ball is let go on a gentle ramp and on a steep ramp of the same length. Which one rolls to the bottom first?
  3. Ramp E goes up 1 square for every 3 squares along. Ramp F goes up 1 square for every 1 square along. Which is steeper?
  4. Why does a ball not roll on a flat table?
  5. Is it easier to walk up a steep hill or a gentle hill? Why?
  6. Why do some mountain roads go in zigzags?
  7. Ramp G goes up 2 for each 1 along. Ramp H goes up 4 for each 2 along. Which is steeper? (Careful!)
  8. Grown-up question: two perfectly slippery slides start at the same height. One is steep, one gentle. Why can you still be going the same speed at the bottom of both?
Answers
  1. The ladder. It goes up a lot for very little along. A wheelchair ramp goes up only a little.
  2. The ball on the steep ramp.
  3. Compare the same "along": in 3 squares along, E goes up 1 and F goes up 3. F is steeper.
  4. Gravity pulls the ball straight down into the table, and none of the pull goes along the table, so nothing starts it rolling.
  5. A gentle hill. On a steep hill you have to lift yourself up a lot with each step. On a gentle hill each step lifts you only a little.
  6. A zigzag road is longer but gentler. Cars and bikes can climb a gentle road more easily and do not go too fast coming down.
  7. They are equally steep! Ramp H goes up 4 for 2 along. If you only go 1 along, it goes up 2, just like G. They are the same steepness, H is just a longer piece.
  8. Your speed at the bottom depends on how far you dropped, and both slides drop the same height. On the gentle slide you speed up more slowly, but for longer, so you end at the same speed. (Rubbing makes real slides a bit different.)

Common mistake

Thinking a bigger ramp is a steeper ramp. A long ramp can be gentle, and a short ramp can be steep. Steepness is about how much it goes up for each step along, not about how big the ramp is. In question 7, the two ramps look different but are equally steep.

Thinking steep always means fast. Steep down is fast, but steep up is slow. Always ask which way you are going.

Where it leads