Problem 49 - Master Resource Book in JEE Main Physics

Question

From a balloon rising vertically upwards at \(5 \mathrm{~m} / \mathrm{s}\) a stone is thrown up at \(10 \mathrm{~m} / \mathrm{s}\) relative to the balloon. Its velocity with respect to ground after \(2 \mathrm{~s}\) is (assume \(\left.g=10 \mathrm{~m} / \mathrm{s}^{2}\right)\) (a) \(\underline{0}\) (b) \(20 \mathrm{~m} / \mathrm{s}\) (c) \(10 \mathrm{~m} / \mathrm{s}\) (d) \(5 \mathrm{~m} / \mathrm{s}\)

Step-by-Step Solution

Answer

The answer is not provided in the given options.

Step 1: Understand the scenario

A stone is thrown from a balloon moving vertically upward. The velocity of the balloon is \(5 \mathrm{~m/s}\) upwards. The stone is given an upward velocity of \(10 \mathrm{~m/s}\) relative to the balloon.

Step 2: Calculate initial velocity with respect to the ground

The initial upward velocity of the stone relative to the ground is the sum of the balloon's velocity and the stone's relative velocity. This gives us:\[v_{i} = 5 \mathrm{~m/s} + 10 \mathrm{~m/s} = 15 \mathrm{~m/s}\] upward.

Step 3: Apply the motion equation

To find the velocity of the stone with respect to the ground after 2 seconds, use the equation \(v = u + gt\), where \(u\) is the initial velocity, \(g\) is the acceleration due to gravity (which is negative as it acts downward), and \(t\) is time. Thus,\[v = 15 \mathrm{~m/s} - 10 \mathrm{~m/s}^{2} \times 2 \mathrm{~s}\]

Step 4: Solve for final velocity

Calculate the value:\[v = 15 \mathrm{~m/s} - 20 \mathrm{~m/s} = -5 \mathrm{~m/s}\]The negative sign indicates that the velocity is downward.

Key Concepts

Kinematics

Kinematics is the branch of physics that describes the motion of objects without considering the causes of motion. It primarily focuses on the concepts of displacement, velocity, and acceleration.
Velocity is the rate of change of position. It's a vector quantity, which means it has both magnitude and direction.
Additionally, we use the kinematic equations to analyze and predict the movement of objects.When analyzing motion:

• Initial velocity (\( u \)): Starting speed and direction of the object.
• Final velocity (\( v \)): Speed and direction at the end of an interval.
• Acceleration (\( a \)): Rate of change of velocity, here influenced by gravity.
• Time (\( t \)): Duration over which the motion occurs.

In the problem, we applied kinematics by calculating the initial velocity relative to the ground and then using the equation \( v = u + at \) to find the stone's velocity after a specific time.

Projectile Motion

Projectile motion is a type of kinematic motion where an object moves along a curved path under the influence of gravity. In this scenario, the stone is thrown upwards from a moving balloon, making it an example of a vertically launched projectile. Key points of projectile motion include:

• Initial Velocity: This is both the balloon's velocity and the stone's relative velocity combined.
• Gravity's Influence: Gravity acts downwardly, accelerating the stone in the opposite direction of the initial throw.
• Motion Analysis: Using the initial velocity and gravitational acceleration, we find the object's velocity at various points in time.

In the problem, the stone's vertical projectile motion is impacted by the combination of its initial velocity and gravity. Using these concepts, we calculated its velocity after 2 seconds to reveal a downward trajectory.

Gravity

Gravity is a fundamental force that attracts two bodies towards each other. On Earth, this force gives weight to physical objects and influences their motion.
When discussing projectile motion, gravity provides a constant acceleration towards the Earth, normally represented as \( g = 10 \mathrm{~m/s^2} \) in simplified problems like this one.Important aspects of gravity in motion include:

• Acceleration due to Gravity: Always acts downward, affecting vertical motion components.
• Impact on Velocity: Gravity decelerates objects moving upwards and accelerates objects moving downwards.
• Trajectory Change: Affects the curvature of projectile paths.

In our scenario, we considered gravity reducing the stone's upward velocity, ultimately causing it to move downward after 2 seconds, shown by a negative resultant velocity.