Problem 59 - Fundamentals of Physics

Question

Water drips from the nozzle of a shower onto the floor \(200 \mathrm{~cm}\) below. The drops fall at regular (equal) intervals of time. the first drop striking the floor at the instant the fourth drop begins to fall. When the first drop strikes the floor, how far below the nocrle are the (a) second and (b) third drops?

Step-by-Step Solution

Answer

(a) 22.1 cm below nozzle; (b) 89.5 cm below nozzle.

Step 1: Understand the problem

We need to determine the positions of the second and third drops when the first drop hits the floor. Since it is given that the first drop strikes the floor at the instant the fourth drop begins to fall, there are three intervals of time between these events.

Step 2: Define time intervals

Since the first drop falls 200 cm, the time taken ( t_f ) can be computed using the equation for the motion under gravity: \( s = \frac{1}{2}gt^2 \). Given:\( s = 200 \, \mathrm{cm} \). Let \( g = 980 \, \mathrm{cm/s^2} \) (approximate value of gravitational acceleration). Solving \( 200 = \frac{1}{2} \times 980 \times t_f^2 \), we get the time taken for the first drop to hit the floor.

Step 3: Calculate time for the first drop to hit the floor

Rearrange the equation from Step 2: \( t_f = \sqrt{\frac{2 \times 200}{980}} \). Simplifying gives: \( t_f = \sqrt{\frac{400}{980}} = \sqrt{0.40816} \approx 0.639 \) seconds. This is the total time from the start to when the first drop hits the floor.

Step 4: Calculate time intervals between drops

Since the first drop takes \( 0.639 \) seconds to reach the floor, and there are three equal-time intervals before the fourth drop falls, let \( t \) be the time interval between drops. Therefore, \( 3t = 0.639 \), and solving gives \( t \approx 0.213 \) seconds.

Step 5: Calculate distance of the second drop

The second drop falls for time \( t \) (0.213 seconds) by the time the first hits the floor. Using the formula \( s = \frac{1}{2}gt^2 \), substitute \( g = 980 \, \mathrm{cm/s^2} \) and \( t = 0.213 \, \mathrm{s} \): \( s = \frac{1}{2} \times 980 \times (0.213)^2 \). Simplifying gives \( s \approx 22.105 \) cm.

Step 6: Calculate distance of the third drop

The third drop has fallen for time \( 2t \) (2 x 0.213 seconds) at the moment the first hits the floor. Therefore, the fall time for the third drop is \( 0.426 \) seconds. Again using \( s = \frac{1}{2}gt^2 \): \( s = \frac{1}{2} \times 980 \times (0.426)^2 \). Simplifying gives \( s \approx 89.544 \) cm.

Key Concepts

Projectile Motion

Projectile motion describes the path an object follows when it is thrown or projected into the air. These motions are influenced mainly by two forces: horizontal velocity and vertical acceleration due to gravity. In a standard projectile motion scenario, the horizontal velocity remains constant, assuming no air resistance, while the vertical velocity is affected by gravity.

For problems involving direct vertical motion, like falling water drops, the horizontal component is negligible. These drops move under the influence of gravity alone. Their vertical motion can be predicted using the formula for motion under gravity:

• The distance fallen: \( s = \frac{1}{2}gt^2 \)
• Where \( s \) is the distance, \( g \) is the acceleration due to gravity \( (980 \text{ cm/s}^2 \text{ in this context}) \), and \( t \) is the time in seconds.

Understanding these basics of projectile motion helps solve and predict the behavior of objects under gravitational influence.

Gravity

Gravity is a natural phenomenon by which all things with mass or energy are brought toward one another. On Earth, it gives weight to physical objects and is measured by the gravitational acceleration value, denoted as \( g \). In this context, gravity's effect on motion is clear, as it pulls objects down toward the Earth.

When you're calculating the motion of falling objects, the usual value of \( g \) in physics problems is approximately \( 9.8 \text{ m/s}^2 \), or \( 980 \text{ cm/s}^2 \). This constant acceleration is fundamental in calculating how fast a falling object like our water drops will hit the ground.

Any object in free fall near Earth's surface will accelerate at this rate, experiencing velocity changes proportional to \( g \). This is why it's important in kinematics, the study of motion, to incorporate \( g \) in understanding and predicting motion.

Motion Under Gravity

Motion under gravity, often referred to as free fall, involves the movement of objects when the only force acting on them is gravity. In the case of the shower drops, it's all about calculating the time they take to hit the ground and the distances they cover.

In free fall:

• The force of gravity causes the object to have a constant acceleration downward.
• There is no initial vertical velocity in our problem since the drop begins to fall from rest.
• The distance and time are connected through equations like: \( s = \frac{1}{2}gt^2 \).

For our shower nozzle drops, each new drop begins its motion at regular intervals, highlighted by the consecutive timing of their falls. By understanding the drop intervals and using the gravitational formula, one can easily track how far each drop has fallen at any given time, reiterating the importance of knowing the basic mechanics of motion under gravity.