Problem 39 - Master Resource Book in JEE Main Physics

Question

A body is fired vertically upwards. At half the maximum height, the velocity of the body is \(10 \mathrm{~ms}^{-1}\). The maximum height raised by the body is \(\begin{array}{ll}\left(g=10 \mathrm{~ms}^{-2}\right) . & \text { [Orissa JEE 2008] }\end{array}\) (a) zero (b) \(10 \mathrm{~m}\) (c) \(15 \mathrm{~m}\) (d) \(20 \mathrm{~m}\)

Step-by-Step Solution

Answer

The maximum height raised by the body is \(20 \mathrm{~m}\).

Step 1: Identify the given information

We are given that the initial velocity of the body at half the maximum height is \(10 \mathrm{~ms}^{-1}\). The acceleration due to gravity \(g\) is \(10 \mathrm{~ms}^{-2}\).

Step 2: Determine the height at half maximum height

Using the equation \(v^2 = u^2 - 2gh\), where \(v = 10 \mathrm{~ms}^{-1}\), \(u = 0 \mathrm{~ms}^{-1}\) at the maximum height, and \(g = 10 \mathrm{~ms}^{-2}\), find the half maximum height \(h/2\).\(10^2 = 0 - 2 \times 10 \times \frac{h}{2}\), solve for \(h/2\).\(100 = -10h\)\(h = 10 \mathrm{~m}\).

Step 3: Calculate the total maximum height

Since \(h = 10 \mathrm{~m}\) represents half of the total height, the total maximum height \(H\) is twice of this value.\(H = 2 \times h = 2 \times 10 = 20 \mathrm{~m}\).

Key Concepts

Kinematics

Kinematics is a branch of physics that deals with the motion of objects without considering the forces that cause such motion. The main focus in kinematics is on the path taken by the object, its velocity, and its acceleration over time. In our exercise, a body is projected vertically upwards, which is a straightforward example of kinematic motion. When the body is at half the maximum height, it still possesses some upward velocity, which gradually decreases due to the force of gravity pulling it back toward the Earth.

Three crucial variables play a role in kinematics:

• Displacement: The change in position of the body, in this case, the vertical distance it travels.
• Velocity: The speed of the body in a specific direction, which decreases as it ascends.
• Acceleration: In our scenario, the constant acceleration due to gravity, which always acts downwards.

By analyzing these variables, especially at different points in its path, we gain insight into different aspects of the motion, such as its maximum height and the time taken to reach it.

Equations of motion

The equations of motion are essential tools in kinematics that help us solve for unknown quantities in problems involving motion. They describe the relationships between displacement, velocity, acceleration, and time. In our exercise, we use one of these equations to find out half the maximum height reached by the body. The key equation used is:
\[ v^2 = u^2 - 2gh \]
Here:

• \(v\) is the final velocity.
• \(u\) is the initial velocity, which is zero at the very peak of the motion.
• \(g\) represents the acceleration due to gravity.
• \(h\) is the height.

This equation helps determine the value of height at any point in the motion. To find the total maximum height, we first calculate the height at which the velocity is 10 m/s using this equation, and then double it to get the full maximum height.

Acceleration due to gravity

Gravity is a natural force that pulls objects toward the center of Earth. In terms of physics, it provides a constant acceleration to any object in free fall. This acceleration is denoted by the symbol \(g\), which typically has a value of \(9.8 \, \text{m/s}^2\) or approximated as \(10 \, \text{m/s}^2\) for simpler calculations, as in our exercise.

The acceleration due to gravity acts downwards, opposite to the direction of an object fired upwards, such as in the exercise scenario. When calculating motion under the influence of gravity:

• Gravity reduces the upward velocity of the body until it reaches its peak point, where the velocity briefly becomes zero.
• After this, the object starts descending due to the constant pull of gravity.
• This acceleration remains constant during the entire motion, which is why it's a crucial factor in kinematic calculations.

Understanding how gravity affects motion is essential for accurately determining various parameters, like the maximum height, in projectile motion exercises.