Problem 5 - Master Resource Book in JEE Main Physics

Question

The height \(y\) and distancex along the horizontal for a body projected in the xy-plane are given by \(y=8 t-5 t^{2}\) and \(x=6 t .\) The initial speed of projection is (a) \(8 \mathrm{~m} / \mathrm{s}\) (b) \(9 \mathrm{~m} / \mathrm{s}\) (c) \(10 \mathrm{~m} / \mathrm{s}\) (d) \((10 / 3) \mathrm{m} / \mathrm{s}\)

Step-by-Step Solution

Answer

The initial speed is 10 m/s, which matches option (c).

Step 1: Understand the equations

We are given two equations: \( y = 8t - 5t^2 \) for vertical motion and \( x = 6t \) for horizontal motion. These equations describe the motion of a projectile along the x and y axes over time \( t \).

Step 2: Recognize initial conditions

The initial conditions, when \( t = 0 \), are \( y(0) = 0 \) and \( x(0) = 0 \). This confirms that the object starts from the origin.

Step 3: Determine initial components

From the equation \( y(t) = 8t - 5t^2 \), the initial vertical velocity coefficient is 8 m/s. From \( x(t) = 6t \), the initial horizontal velocity coefficient is 6 m/s.

Step 4: Calculate initial speed

The initial speed of the projection \( v_0 \) is the vector sum of the initial velocities in the x and y directions. Calculate it using: \[ v_0 = \sqrt{(v_{0x})^2 + (v_{0y})^2} \]Substitute \( v_{0x} = 6 \) m/s and \( v_{0y} = 8 \) m/s:\[ v_0 = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ m/s} \]

Step 5: Choose the correct answer

Compare the calculated initial speed (10 m/s) with the given options. The correct choice based on our calculations is (c) 10 m/s.

Key Concepts

Initial Velocity

In projectile motion, the initial velocity is the speed a projectile has when it is first launched into the air. This velocity can be split into two components: a horizontal component and a vertical component. This is crucial because each component affects how the projectile moves in those directions over time.

The horizontal component, typically represented by \( v_{0x} \), remains constant if we assume no air resistance. In our problem, this initial horizontal velocity is 6 m/s, derived from the function \( x(t) = 6t \).

The vertical component, represented by \( v_{0y} \), is initially influenced by gravitational acceleration soon after being launched. In the given function \( y(t) = 8t - 5t^2 \), the coefficient before \( t \) is the initial vertical velocity, which is 8 m/s.

Understanding these components will help us analyze the projectile's overall motion and determine its initial velocity when launched.

Vector Sum

When dealing with physics problems involving vectors, the concept of a vector sum becomes essential. Vectors are quantities that have both magnitude and direction, such as velocity in projectile motion. To find the initial speed of the projectile, we need to combine the horizontal and vertical velocity components into a single vector.

To calculate this, we use the Pythagorean theorem as we assume these components form a right triangle. The formula used is:

• \[ v_0 = \sqrt{(v_{0x})^2 + (v_{0y})^2} \]

In this context:

• \( v_{0x} = 6 \) m/s
• \( v_{0y} = 8 \) m/s

Substituting, we get:

• \[ v_0 = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ m/s} \]

This calculation confirms that the initial velocity of our projectile is 10 m/s.

Equations of Motion

The equations of motion play a significant role in solving problems related to projectiles, allowing us to predict their future position and velocity at any given time. They are the mathematical descriptions of an object's motion within a force field, like gravity.

In our exercise, we use two separate equations to describe movement:

• \( y(t) = 8t - 5t^2 \) describes vertical motion, with gravity affecting it as represented by the \(-5t^2\), where \( -5\) is half the acceleration due to gravity (as it considers time squared).
• \( x(t) = 6t \) describes horizontal motion, which is unaffected by gravity, maintaining constant speed due to the absence of horizontal forces.

These equations help us determine where the projectile is at any time \( t \), and the initial velocities, which are essential for calculating the overall initial velocity of the projectile through the vector sum.