C. Energy and Momentum · Work, Energy and Simple Harmonic Motion

Springs in Energy Problems: Launchers, Drops and Bungee Cords

C2.3 C3.1 C2.2About 75 minutes

Learning goals

I can write and use the full energy equation with kinetic, gravitational and elastic energy and friction.
I can find speeds and maximum compressions in spring-launcher and spring-drop problems.
I can solve the quadratic equations that arise when gravity and a spring act together.
I can explain where an object on a spring or bungee cord is moving fastest.

Words to know

mechanical energy (with springs)
Em = Ek + Eg + Ee; conserved when only gravity and ideal spring forces do work.

maximum compression
The greatest distance a spring is squeezed, where the object is momentarily at rest.

bungee cord
An elastic cord that stretches only after its natural (unstretched) length is reached; often modelled as an ideal spring.

quadratic formula
x = [−b ± √(b² − 4ac)] ÷ (2a), used to solve ax² + bx + c = 0.

Putting springs into the energy equation

Pinball launchers, pogo sticks, car bumpers and bungee jumps all mix three kinds of mechanical energy: kinetic, gravitational and elastic. Now that you have Ee = ½kx², you can track all three at once. The idea is the same as before: list the energies at the start and the end, include any thermal energy from friction, and solve.

Because the spring force and gravity are both conservative, the mechanical energy

Em = Ek + Eg + Ee

is conserved when no friction acts. With friction:

Ek₁ + Eg₁ + Ee₁ = Ek₂ + Eg₂ + Ee₂ + E_th

A strategy for spring problems

Sketch the situation at the key moments: release, leaving or touching the spring, maximum compression or stretch, highest point.
For each moment write the speed, the height (from a reference level you choose) and the spring's stretch or compression x.
Write the energy equation between the two moments the question is about.
If the unknown appears as both x and x² (common when gravity and a spring both act), you will get a quadratic equation: use the quadratic formula and keep the positive root that makes sense.

Worked example

Example 1: A launcher on a rough table

A 0.50 kg block is pressed against a spring (k = 200 N/m), compressing it 0.10 m, on a table with μ_k = 0.20. It is released and leaves the spring when the spring reaches its natural length. (a) How fast is the block moving when it leaves the spring? (b) How far from its release point does it stop?

(a) Ee₁ = ½(200)(0.10)² = 1.0 J. Friction: F_f = 0.20(0.50)(9.8) = 0.98 N, so E_th over 0.10 m = 0.098 J.
Ek₂ = Ee₁ − E_th = 1.0 − 0.098 = 0.902 J. v = √(2 × 0.902 ÷ 0.50) = 1.9 m/s.
(b) All 1.0 J ends up as thermal energy: 0.98d = 1.0, so d = 1.02 m from the release point (0.92 m after leaving the spring).

(a) 1.9 m/s; (b) about 1.0 m from the release point.

Worked example

Example 2: A block dropped onto a spring

A 2.0 kg block is dropped from rest 0.50 m above the top of a vertical spring (k = 1000 N/m). Find the maximum compression of the spring.

Choose the reference level at the lowest point (maximum compression x). Then the block falls a total height of 0.50 + x.
At the start: Eg = mg(0.50 + x), Ek = 0, Ee = 0. At the lowest point: Ek = 0 (momentarily at rest), Eg = 0, Ee = ½kx².
mg(0.50 + x) = ½kx² → 19.6(0.50 + x) = 500x² → 500x² − 19.6x − 9.8 = 0.
x = [19.6 + √(19.6² + 4(500)(9.8))] ÷ (2 × 500) = (19.6 + 141.4) ÷ 1000 = 0.161 m. (The negative root has no meaning here.)

The spring compresses about 16 cm.

The three key moments of Example 2. Measuring heights from the lowest point means the block falls 0.50 m + x in total, and all of that gravitational energy ends up in the spring.

Worked example

Example 3: A bungee jump

A 60 kg jumper steps off a platform attached to a bungee cord with an unstretched length of 15 m. Model the cord as an ideal spring with k = 200 N/m, and ignore air resistance. (a) How far below the platform is the lowest point? (b) Where is the jumper moving fastest, and how fast?

(a) Let x be the cord's stretch at the lowest point. The jumper falls 15 + x. Energy: mg(15 + x) = ½kx².
588(15 + x) = 100x² → 100x² − 588x − 8820 = 0 → x = [588 + √(588² + 4(100)(8820))] ÷ 200 = (588 + 1968.2) ÷ 200 = 12.78 m.
Lowest point: 15 + 12.78 = 27.8 m below the platform.
(b) The jumper speeds up as long as gravity is larger than the cord's pull, so the speed is greatest where kx = mg: x = 588 ÷ 200 = 2.94 m, that is 17.94 m below the platform.
Energy from the start to that point: ½mv² = mg(17.94) − ½k(2.94)² = 10 549 − 864 = 9684 J (keeping all digits: v² = 351.6 − 28.8 = 322.8), so v = 18 m/s (17.97 m/s).

(a) About 28 m below the platform; (b) fastest 17.9 m below the platform, at about 18 m/s. Real bungee operations use cords tested and rated for each jumper's mass, plus large safety margins.

Energy during the bungee jump in Example 3 (reference level at the lowest point). The total bar height never changes: Eg becomes Ek, and then Ek and the rest of Eg become Ee in the cord.

Why the fastest point is not where the cord goes tight

Many students think the jumper is fastest when the cord starts to stretch. But just after that moment, the cord's upward pull (kx) is still smaller than the weight, so the net force is still downward and the jumper keeps speeding up. Speeding up stops only when kx = mg. The same is true for a block dropped onto a spring: it is fastest slightly below the point where it first touches the spring.

Just for fun

Why did the block refuse to leave the spring?

It was too attached, and every time it tried to move on, something pulled it right back.

Hands-on activity

Bounce a ball onto a sponge spring

You need: a small ball that bounces well (a tennis ball works), a kitchen sponge (it acts as a soft spring), a stack of books, a ruler, a tape measure and a kitchen scale.

Weigh the ball. Stand the ruler on the table next to the sponge.
Find the sponge's k: press a book of known mass on it and measure the compression x; k ≈ mg ÷ x. (Stack two or three books for a better reading.)
Drop the ball onto the sponge from 30 cm, 60 cm and 90 cm above the sponge's top. Film the bounce in slow motion and estimate the largest compression.
Use mg(h + x) = ½kx² to predict the compression for each drop height, and compare.

What to look for: compression grows with drop height, but more slowly than the height (roughly like its square root). Your predictions will not be perfect because a sponge is not an ideal spring and some energy becomes thermal energy, but the trend should match.

Your turn

A 0.20 kg puck is pushed against a spring (k = 250 N/m), compressing it 8.0 cm, on frictionless ice. How fast is the puck moving when it leaves the spring?

A 1.5 kg block sliding at 2.0 m/s on a frictionless surface hits a spring (k = 600 N/m). Find the maximum compression.

In the last question, how fast is the block moving when the spring is compressed 5.0 cm?

A vertical toy launcher has a spring (k = 400 N/m) compressed 10 cm. It fires a 50 g ball straight up. How high does the ball rise above its starting (compressed) position? Ignore air resistance.

A block oscillates on a horizontal frictionless spring. At which position is its speed greatest?

Write the full energy equation for a system with gravity, a spring and friction.

More practice: warm-up

List the three forms of mechanical energy in spring problems.

At maximum compression, what is the speed of an object on a spring?

When a horizontal spring returns to its natural length, what is Ee?

A spring stores 3.0 J and launches a block on a frictionless surface. What is the block's Ek when it leaves the spring?

Why can a spring-and-gravity problem lead to a quadratic equation?

More practice: on your own

A 2.0 kg block is dropped from 0.30 m above a vertical spring (k = 800 N/m). Find the maximum compression.

A 0.40 kg block is pressed against a spring (k = 300 N/m), compressing it 12 cm, on a floor with μ_k = 0.30. It is not attached to the spring. How far from its release point does it stop?

A pinball plunger spring (k = 120 N/m) is compressed 6.0 cm and launches a 0.080 kg ball up a table inclined at 6.0°. Ignoring friction, how fast is the ball moving after travelling 1.0 m along the table from its release point?

A 70 kg jumper uses a bungee cord 20 m long with k = 150 N/m. How far below the platform is the lowest point?

A 1000 kg car moving at 2.0 m/s hits a spring-type bumper with k = 4.0 × 10⁵ N/m. Find the maximum compression and the maximum force on the car.

Challenge

A 0.50 kg block starts from rest and slides 1.0 m down a frictionless 30° incline, then hits a spring (k = 200 N/m) lying along the incline. Find the maximum compression.

A 0.20 kg block on a frictionless horizontal surface is attached to a spring (k = 50 N/m). It is pulled 0.10 m from equilibrium and released. Find its speed when the spring is stretched 0.060 m.

A bungee operator wants an 80 kg jumper's lowest point to be 30 m below the platform, using a cord 12 m long. What spring constant should the cord have (ideal-spring model)?

Check your understanding

Quiz

A block slides on a frictionless surface into a spring and compresses it. At maximum compression:

A spring (k = 800 N/m) compressed 0.050 m launches a 0.25 kg block on a frictionless table. The block's speed is:

A bungee jumper is moving fastest when:

A block is dropped onto a vertical spring. Choosing the reference level at the point of maximum compression is useful because:

For a spring launcher on a rough table, the energy equation from release to leaving the spring is:

Total mechanical energy Ek + Eg + Ee is conserved when:

For parents and teachers

Big idea (C2.3, C3.1): kinetic, gravitational and elastic energy can all change in one problem; with no friction their total is constant. Students choose key moments, write the energies at each, and solve, sometimes with a quadratic equation.

How to help: watch a video of a bungee jump or a pogo stick together and ask your teen to narrate where the energy is at each moment. Ask where the jumper is fastest (not where the cord goes tight).

Common difficulties:

Forgetting that a block dropped on a spring keeps falling while it compresses the spring (total drop is h + x).
Choosing different reference levels at different moments.
Discarding the wrong root of the quadratic.
Treating a bungee cord as stretched from the moment of the jump (it only stretches after its natural length).