Question

Raindrops fall to the ground from a cloud \(1700 \mathrm{~m}\) above Earth's surface. If they were not slowed by air resistance, how fast would the drops be moving when they struck the ground? Would it be safe to walk outside during a rainstorm?

Short answer

The raindrops would be moving at a speed of approximately 183 m/s when they strike the ground. It would not be safe to walk outside during a rainstorm if raindrops were falling at these speeds.

Step-by-step solution

Identify Knowns and Unknowns

In this situation, three quantities are known: the initial speed \(u = 0 \; \mathrm{m/s}\) (because the raindrops start from rest), the acceleration \(g = 9.8 \; \mathrm{m/s^2}\) (average acceleration due to Earth's gravity), and the dislocation \(s = 1700 \; \mathrm{m}\). The aim is to find the unknown variable, the final speed \(v\).

Choosing the Relevant Equation

The equation \(v^2 = u^2 + 2gs\), which derives from the equations of motion, will be used to calculate the final speed \(v\). This equation connects the final speed with the initial speed, the acceleration due to gravity and the distance covered, all of which are known quantities.

Substituting Known Values into the Equation

Substitute all known values into the chosen equation: \(v^2 = 0 + 2 * 9.8 * 1700\). Therefore, \(v^2 = 33320\).

Solve for the Unknown Variable

Isolate the variable \(v\) by taking the square root of both sides: \(v = \sqrt{33320}\). Therefore, \(v \approx 183 \; \mathrm{m/s}\). This is the final speed of the raindrops when they strike the ground.

Analyze the Safety of Walking outside during a Rainstorm

The calculated speed is very high, a raindrop moving at approximately 183 m/s is moving at a speed faster than most commercial jet planes. Therefore, walking outside in such a rainstorm would be extremely dangerous.

Key concepts

Acceleration Due to Gravity

When discussing raindrops falling from a height, the concept of acceleration due to gravity is fundamental. On Earth, this acceleration is approximately \(9.8 \, \mathrm{m/s^2}\). This constant represents how fast an object speeds up when it is in free-fall. It means that, for each second an object is falling, its speed increases by \(9.8 \, \mathrm{m/s}\) if no other forces are acting on it.

This value is crucial for calculations involving free-falling objects, as it allows us to predict how fast they will be moving after falling for a certain amount of time or distance. In our raindrop example, although the raindrops start from rest, gravity pulls them down towards the Earth, increasing their speed steadily until they reach the ground. Without air resistance, we can use this acceleration to calculate how fast they'd be moving upon impact.

Initial Speed

Initial speed is a term used to describe the speed at which an object starts its motion. For the raindrops in the exercise, the initial speed \(u\) is \(0 \, \mathrm{m/s}\) because they begin their journey from rest in the cloud. This means, before gravity acts on them, they are stationary.

Recognizing the initial speed as zero simplifies our calculations, as any change in speed is solely due to the acceleration caused by gravity. This is a key consideration in using the equations of motion effectively, as our initial speed significantly impacts the calculation of the final speed over a given distance.

Final Speed

Final speed is the speed at which an object is moving at the end of its motion. In the case of the raindrops, this is the speed at which they hit the ground. By knowing the initial speed and the acceleration due to gravity, we can apply the equations of motion to determine this final speed.

The relevant equation here is \(v^2 = u^2 + 2gs\), which directly links the known quantities to find the final speed \(v\). By substituting the values:

• \(u = 0 \, \mathrm{m/s}\)
• \(g = 9.8 \, \mathrm{m/s^2}\)
• \(s = 1700 \, \mathrm{m}\)

we find that \(v\) is approximately \(183 \, \mathrm{m/s}\). Such speed underlines the importance of understanding these concepts, as it emphasizes how quickly relatively small objects can travel due to gravity alone, without any other forces in play.

Air Resistance in Physics

While our calculations assumed no air resistance, it is rare for objects on Earth to fall without facing this force. Air resistance acts in the opposite direction of motion, reducing the acceleration and eventually reaching a terminal velocity where the net acceleration becomes zero.

This resistance plays a crucial role in the natural world, significantly affecting how fast falling objects like raindrops eventually travel. With air resistance, raindrops typically reach a much safer terminal velocity of about \(9 \, \mathrm{m/s}\), allowing us to walk in the rain safely. Understanding this force is critical in real-world applications, ensuring predictions and safety assessments are accurate in everyday situations.