Problem 53 - Master Resource Book in JEE Main Physics

Question

Water drops fall from a tap on the floor \(5 \mathrm{~m}\) below at regular intervals of time, the first drop striking the floor when the fifth drop begins to fall. The height at which the third drop will be, from ground, at the instant when first drop strikes the ground, will be \(\left(g=10 \mathrm{~ms}^{-2}\right)\) (a) \(1.25 \mathrm{~m}\) (b) \(2.15 \mathrm{~m}\) (c) \(2.73 \mathrm{~m}\) (d) \(3.75 \mathrm{~m}\)

Step-by-Step Solution

Answer

1.25 m

Step 1: Determine Time for a Drop to Hit Ground

First, calculate the time it takes for a drop to travel 5 meters to the ground under gravity. Use the equation for free fall: \[ s = \frac{1}{2} g t^2 \]Given, \( s = 5 \) m and \( g = 10 \) m/s². Substituting gives:\[ 5 = \frac{1}{2} \times 10 \times t^2 \]\[ t^2 = 1 \]\[ t = 1 \text{ second} \]

Step 2: Calculate Time Interval Between Drops

The fifth drop begins to fall when the first drop hits the ground, indicating that there are four intervals between the five drops during the time the first drop falls. Hence, each interval, \( \Delta t \), is:\[ \Delta t = \frac{1}{4} \times 1 = 0.25 \text{ seconds} \]

Step 3: Determine Time Elapsed for Third Drop

The third drop begins to fall two intervals after the first drop. Thus, the time elapsed after the third drop starts falling, by the time the first drop hits the ground, is:\[ t_3 = 1 - 2 \times 0.25 = 0.5 \text{ seconds} \]

Step 4: Calculate Height of Third Drop

Use the free fall equation to find the height of the third drop from the ground after 0.5 seconds:\[ s_3 = \frac{1}{2} \times 10 \times (0.5)^2 \]\[ s_3 = 1.25 \text{ meters} \]Thus, the third drop is 1.25 meters from the ground.

Key Concepts

Free Fall

Free fall describes the motion of objects moving solely under the influence of gravity. In this scenario, we assume negligible air resistance, allowing objects to experience uniform acceleration towards the Earth. This acceleration is denoted as the gravitational constant, \( g \), which is typically valued at \( 9.8 \ \text{m/s}^2 \), but often approximated as \( 10 \ \text{m/s}^2 \) for simplification. When an object is in free fall, the only force acting on it is gravity, leading to its continual acceleration as it descends.

• Objects in free fall will cover more distance with each passing second because their speed increases continually.
• The key equation governing free fall is \( s = \frac{1}{2} g t^2 \), indicating that the distance \( s \) traveled is proportional to the square of the time \( t \) in free fall.

If you drop an object, it begins with an initial velocity of zero and gains velocity as it falls, leading it to move faster the longer it falls.

Equations of Motion

The equations of motion are essential tools in kinematics. They help us describe the movement of objects and predict future positions and velocities. There are three key kinematic equations, each helping with different scenarios. When dealing with free fall and vertical motion, the most common equation is \( s = \frac{1}{2} g t^2 \). This equation gives the vertical distance \( s \), given the acceleration due to gravity \( g \) and time \( t \).

• These equations allow us to calculate various unknowns, like the time of flight or height at a given point of time.
• An understanding of these equations enables us to solve complex motion problems step-by-step.

With these equations, we can also explore how changing variables affect motion. For example, we see how the interval between water drops can be calculated based on the total time of fall and the number of intervals.

Time of Flight

The time of flight in kinematics refers to the duration of time an object is in motion from the point of release until it reaches the ground. Understanding the time of flight is crucial for predicting and analyzing the trajectory and impacts of falling objects. In our tap water drop problem, the first drop's time of flight is exactly 1 second due to our calculated scenario.

• The time of flight is influenced by initial speed, height from which the object is dropped, and gravitational forces at play.
• In a sequence like the water drop problem, understanding the time intervals between successive drops helps us predict their positions over time.

By segmenting the overall time into intervals, as shown in the problem, we can identify when each drop falls and calculate their respective heights from the ground at any given moment.