What if a slide were steeper?

You climb up and sit at the top of the slide. Then whoosh, down you go!

Some slides are gentle. Some are steep. What if a slide were much steeper?

A slide is a ramp: a sloping surface that lets gravity carry you down. On a gentle slope you glide down slowly, and on a very gentle one you might not move at all. On a steeper slope you rush down fast.

But how much faster, and why? In this lab a wooden block on a wooden ramp stands in for a child on a slide. The ramp is always 4 meters long. You can tilt it from 30° to 45° and measure how quickly the block speeds up, how fast it is going at the bottom and how far it slides across the floor.

OpenStax treats a slope as an inclined plane, whether it is a ramp to a loading dock or a mountain under a skateboarder: the basic physics is the same. The weight splits into a component along the slope, which drives the motion, and a component into the surface, which sets the normal force and so the friction.

This lab uses the friction-slide model with wood on wood (μs = 0.5, μk = 0.3, OpenStax's Table 5.1 values), a 4 m ramp and a block released from rest. It raises the angle from 30° to 45° and compares the acceleration, the speed at the bottom and the time to get there.

First, the ramp is gently tilted. The block slides down.

It speeds up as it goes. Then it slides across the floor and stops.

In the model, tilt the 4-meter wooden ramp to 30° and let the block go. It starts to slide, speeding up by about 2.4 meters per second every second. It reaches the bottom after about 1.8 seconds, moving at about 4.3 meters per second, then slides about 3.2 meters across the wooden floor before friction stops it.

Try a gentler slope, 26°. The block does not move at all! On wood, friction can hold it still on any slope up to about 26.6°.

At 30° the model gives a = g(sin θ − μk cos θ) = 9.81(0.500 − 0.3 × 0.866) = 2.36 m/s². Over L = 4 m that yields v_b = √(2aL) = 4.34 m/s after t = √(2L/a) = 1.84 s, and a floor slide of v_b²/(2μk g) = 3.20 m.

The block starts from rest only if tan θ > μs, so the threshold is arctan 0.5 = 26.6°: at 26° the model reports "stays put", at 27° it slides with a = 1.83 m/s². Mass cancels from every result; the model gives the same numbers for 5 kg and 40 kg blocks.

Now the ramp is much steeper. Whoosh! The block zooms down.

It gets to the bottom sooner, and it is going faster. It slides much farther across the floor too.

Now tilt the ramp to 45°. The block speeds up about twice as quickly as before, about 4.9 meters per second every second. It reaches the bottom in only about 1.3 seconds, moving at about 6.2 meters per second, and then slides about 6.6 meters across the floor, more than twice as far.

The ramp is the same length, so a steeper ramp is also a taller ramp. At 30° its top is 2 meters up; at 45° it is about 2.8 meters up.

At 45° the model gives a = 9.81(0.707 − 0.3 × 0.707) = 4.86 m/s², 2.06 times the 30° value, even though sin θ rose only by a factor of 1.41. The friction term falls (cos θ shrinks) while the driving term grows. Results: v_b = 6.23 m/s, t = 1.28 s, floor slide 6.60 m.

With a fixed length the height also grows, from L sin 30° = 2.0 m to 2.83 m. At the model's maximum, 60°, a = 7.02 m/s², v_b = 7.50 m/s and the block slides 9.55 m across the floor.

Gravity pulls the block down. On a steep ramp, more of that pull points down the ramp.

And friction holds on less. So the block goes faster!

Gravity always pulls straight down. On a ramp, only part of that pull points along the slope; the rest presses the block into the ramp. The steeper the ramp, the more of the pull points down the slope.

Friction depends on how hard the block is pressed into the surface. On a steeper ramp the block is pressed in less, so friction holds it back less. More pull and less friction: that's why the block speeds up so much faster. Our model uses typical textbook friction values for wood and ignores air resistance.

Resolving the weight: mg sin θ along the slope, mg cos θ into it. OpenStax notes that simple friction is proportional to the normal force, here mg cos θ, so kinetic friction is μk mg cos θ. Net force: mg(sin θ − μk cos θ); m cancels. As θ grows, the first term rises and the second falls.

Static friction is usually greater than kinetic friction, OpenStax adds, which is why a block can stay put up to arctan μs and then, once moving, keep accelerating. Model limits: Coulomb friction independent of speed and contact area, no tipping or rolling, a smooth corner onto the floor, and no air resistance.

A safety guide says playground slides should not be too steep.

The bottom of a slide should be nearly flat. It helps you keep your balance and stand up.

Real playground slides are kept fairly gentle. The U.S. Consumer Product Safety Commission's playground safety handbook recommends that slides for preschool and school-age children slope, on average, no more than 30°, with no part steeper than 50°. Slides for toddlers should average no more than 24°.

The handbook also asks for a nearly flat exit at the bottom, between level and a 4° slope, to help children keep their balance as they go from sitting to standing. That flat part works like the floor in our model, where friction slows the slider down.

The CPSC Public Playground Safety Handbook recommends that, for preschool- and school-age children, the average incline of a slide chute be no more than 30° (a height-to-horizontal-length ratio of 0.577) with no section steeper than 50°; for toddlers, an average of no more than 24° (ratio 0.445) and no section over 30°. The chute exit region should be between 0 and −4° from horizontal.

The 30° average is this lab's starting slope. Galileo used ramps the other way, to slow motion down: in Michael Fowler's University of Virginia lecture, Galileo reports that times of descent along his channel at various inclinations followed the ratios he had predicted.