Energy and Speed on a Water Slide: A Deep Dive Into Conservation of Energy

An exciting scenario to consider is a child (mass 30 kg) who starts their adventure at the top of a water slide, with a height of 6.0 meters. Assuming a frictionless environment, how fast will they be traveling at the bottom of the slide? This can be analyzed using the principle of conservation of energy, where potential energy at the top is converted to kinetic energy at the bottom.

Calculating Potential Energy

The first step in this analysis is to calculate the potential energy (PE) at the top of the slide using the formula:

PE mgh

Where:

m 30 kg (mass of the child) g 9.81 m/s2 (acceleration due to gravity) h 6.0 m (height of the slide)

Substitute the values into the formula:

PE 30 kg × 9.81 m/s2 × 6.0 m

PE 1764.6 J (joules)

Converting Potential Energy to Kinetic Energy

At the bottom of the slide, all potential energy is converted to kinetic energy (KE). The formula for kinetic energy is:

KE 0.5mv2

Setting the potential energy equal to the kinetic energy at the bottom:

PE KE

1764.6 J 0.5 × 30 kg × v2

Solving for v:

Rearranging the equation:

1764.6 15v2

v2 1764.6 / 15

v2 117.64

v √117.64 ≈ 10.84 m/s

Alternative Calculation

For a more straightforward approach, we can use the simplified formula for velocity when an object falls straight down:

V √(2gh)

V √(2 × 9.81 m/s2 × 6.0 m)

V √117.72 ≈ 10.85 m/s

This value is nearly identical to the one calculated above, confirming the principle of conservation of energy.

Conclusion

Understanding the relationship between potential and kinetic energy allows us to predict the speed of the child at the bottom of the water slide. This analysis not only applies to amusement park rides but also to many real-world physics problems.