Problem 4 - Master Resource Book in JEE Main Physics

Question

An aeroplane is flying in a horizontal direction with a velocity \(600 \mathrm{kmh}^{-1}\) at a height of \(1960 \mathrm{~m}\). When it is vertically above the point \(A\) on the ground, a body is dropped from it. The body strikes the ground at point \(B\). Calculate the distance \(A B\). (a) \(3.33 \mathrm{~km}\) (b) \(333 \mathrm{~km}\) (c) \(33.3 \mathrm{~km}\) (d) \(3330 \mathrm{~km}\)

Step-by-Step Solution

Answer

The distance AB is (a) 3.33 km.

Step 1: Understand the problem

An aeroplane is flying horizontally with a speed of 600 km/h. A body is dropped from it when it's vertically above a point A, at a height of 1960 meters. We need to find the horizontal distance AB between point A, where it was dropped, and point B, where it hits the ground.

Step 2: Convert velocity units

The velocity of the aeroplane is given as 600 km/h. We need to convert this into meters per second for consistency with the height given in meters. Using the conversion: 1 km/h = 1000/3600 m/s, we have: \[600 \text{ km/h} = 600 \times \frac{1000}{3600} \text{ m/s} = \frac{600 \times 1000}{3600} \text{ m/s} \approx 166.67 \text{ m/s}\]

Step 3: Calculate time taken to reach the ground

Use the formula for the time of flight in free fall, where the initial vertical velocity is zero and the only force is gravity (\(g = 9.8 \text{ m/s}^2\)). The height given is 1960 meters, so using the formula: \[ t = \sqrt{\frac{2h}{g}} \]Substitute the values: \[ t = \sqrt{\frac{2 \times 1960}{9.8}} \approx \sqrt{400} = 20 \text{ seconds} \]

Step 4: Calculate the horizontal distance

Now that we know the time taken to fall, we can find the horizontal distance using the formula: \[ d = v \times t \]where \( v \) is the horizontal velocity (166.67 m/s), and \( t \) is the time (20 seconds).Substitute the values: \[ d = 166.67 \times 20 = 3333.4 \text{ meters} \]Converting to kilometers: \[ 3333.4 \text{ meters} = 3.3334 \text{ kilometers} \]

Step 5: Choose the correct option

Compare the calculated distance with the given options. The closest value to 3.3334 km is 3.33 km. Hence, the correct answer is (a) 3.33 km.

Key Concepts

Horizontal Motion

Horizontal motion refers to movement along a straight line on a horizontal plane. In our exercise, the airplane is moving in the horizontal direction, maintaining a constant velocity of 600 km/h. This implies that the speed does not change, as no forces act horizontally to accelerate or decelerate it. This consistency is crucial when calculating how far horizontally the dropped object will travel since the only factor affecting the horizontal distance is time. Once the object is released, it continues to travel horizontally at the same speed of the airplane.

• This is possible because horizontal motion is independent of vertical motion, meaning the horizontal velocity remains constant.
• In problems involving an object dropped from a moving vehicle, the horizontal motion of the object will match that of the vehicle until it impacts the ground.
• The calculation of horizontal distance relies solely on the horizontal speed and the time it takes for the object to hit the ground.

Free Fall

Free fall describes the motion of an object under the influence of gravitational force alone. In this scenario, once the body is released from the aeroplane, it is solely influenced by gravity and starts accelerating downwards with an acceleration of 9.8 meters per second squared (m/s²). The body had no initial vertical velocity.
To calculate how long it takes for the object to hit the ground, the formula for time of flight when an object is in free fall is used: \[ t = \sqrt{\frac{2h}{g}} \]where \(h\) is the height (1960 m) from which the body is dropped, and \(g\) is the gravitational acceleration (9.8 m/s²). This allows us to find the time it takes for the body to fall and reach the ground.

• Free fall outcomes are independent of the object’s horizontal velocity;
• The only factor influencing the fall is gravitational force;
• Understanding free fall is crucial for calculating the total time the object remains airborne.

Conversion of Units

Working with a mix of units necessitates converting values to a common measurement system to ensure calculations are accurate. Typically, the international standard for equations of physics is to use meters, seconds, and kilograms.
In the given problem, the airplane's velocity is initially provided in kilometers per hour (km/h). For consistency with the height measured in meters, we must convert this speed to meters per second (m/s). The conversion factor is:

• 1 km/h equals 1000/3600 m/s, which accounts for the 1000 meters in a kilometer and the 3600 seconds in an hour.

Applying this conversion, the airplane's velocity becomes 166.67 m/s. This ensures that when calculating further attributes of motion, such as distance traveled horizontally or time.
• Unit conversion helps maintain numerical consistency;
• Avoids errors that come from inconsistent units across measurements.

Time of Flight

The time of flight is the total time an object spends in the air from the moment it is released until it touches the ground. For objects in free fall from a given height, we calculate the time of flight using the formula \[ t = \sqrt{\frac{2h}{g}} \].
In our case, with the height of 1960 meters and gravitational acceleration \(g = 9.8 \text{ m/s}^2\), we find that the time it takes for the object to reach the ground is 20 seconds.

• The time of flight is critical in determining how long the object travels due to horizontal velocity before it impacts the ground.
• For an object dropped from a height, despite its horizontal motion, the time of flight depends entirely on vertical motion.

Understanding time of flight is essential to comprehending how different axes of motion function independently and concurrently during a projectile's journey.