Problem 3 - Master Resource Book in JEE Main Physics

Question

A bomber plane moves horizontally with a speed of \(500 \mathrm{~ms}^{-1}\) and a bomb releases from it, strikes the ground in 10 s. Angle at which it strikes the ground will be \(\left(g=10 \mathrm{~ms}^{-2}\right)\) (a) \(\tan ^{-1}\left(\frac{1}{5}\right)\) (b) \(\tan \left(\frac{1}{5}\right)\) (c) \(\tan ^{-1}(1)\) (d) \(\tan ^{-1}(5)\)

Step-by-Step Solution

Answer

(a) \(\tan^{-1}\left(\frac{1}{5}\right)\)

Step 1: Determine the Vertical Velocity of the Bomb

Using the formula for an object in free fall, the final vertical velocity \( v_y \) is given by \( v_y = g \times t \), where \( g = 10 \text{ ms}^{-2} \) and \( t = 10 \text{ s} \). Substituting the values, we get \( v_y = 10 \times 10 = 100 \text{ ms}^{-1} \).

Step 2: Analyze Horizontal Movement

Since the plane moves at a constant speed, the horizontal velocity \( v_x \) of the bomb remains the same as the horizontal velocity of the plane, which is \( v_x = 500 \text{ ms}^{-1} \).

Step 3: Calculate the Angle of Impact

The angle \( \theta \) at which the bomb strikes the ground is given by the tangent of the angle between the velocity components \( \tan(\theta) = \frac{v_y}{v_x} \). Substituting the values, we have \( \tan(\theta) = \frac{100}{500} = \frac{1}{5} \).

Step 4: Determine the Angle Using Inverse Tangent

Now compute the angle \( \theta \) as \( \theta = \tan^{-1}\left(\frac{1}{5}\right) \).

Key Concepts

Horizontal Velocity

Horizontal velocity in projectile motion refers to the constant speed at which an object moves along the horizontal axis. In our scenario, when the bomber plane drops the bomb, it continues to move horizontally with the velocity of the plane. This speed remains unaltered by gravity, making horizontal movement predictable and constant. In our specific example, the horizontal velocity is given as \( 500 \, \text{ms}^{-1} \).

This constant speed means that:

• The horizontal velocity does not change over time until another force acts upon it.
• It allows us to track the horizontal distance traveled over any time frame by using the simple formula \( \text{distance} = \text{velocity} \times \text{time} \).

Understanding horizontal velocity is crucial, as it contributes to figuring out where a projectile will land relative to its starting point.

Vertical Velocity

Vertical velocity in projectile motion reflects the movement of an object along the vertical axis, typically influenced by gravity. When the bomb is released from the plane, it starts accelerating downward under gravity's pull. This adds a vertical component to the movement, unlike the horizontal component which remains constant. In this exercise, we know gravity \( g \) is \( 10 \, \text{ms}^{-2} \), and the time is \( 10 \, \text{s} \).

To calculate the final vertical velocity, we use the equation:

\( v_y = g \times t = 10 \times 10 = 100 \, \text{ms}^{-1} \).

• The vertical velocity increases over time as gravity continuously acts on the object.
• This velocity changes throughout the fall, unlike horizontal velocity which stays the same.
• Knowing the final vertical speed helps in determining other aspects, like the angle of impact with the ground.

Angle of Impact

The angle of impact describes the angle at which a projectile hits the ground. It's a critical component for understanding the full trajectory of a projectile. The angle is determined by both horizontal and vertical velocities at the time of impact. In our exercise, these velocities are \( v_x = 500 \, \text{ms}^{-1} \) for horizontal and \( v_y = 100 \, \text{ms}^{-1} \) for vertical.

To find the angle \( \theta \), we use the tangent function, where:
\( \tan(\theta) = \frac{v_y}{v_x} = \frac{100}{500} = \frac{1}{5} \).

Then we apply the inverse tangent to find the angle:
\( \theta = \tan^{-1}\left(\frac{1}{5}\right) \).

• This calculation provides the angle between the trajectory path and the horizontal line when the bomb hits the ground.
• The angle of impact is valuable in practical scenarios, such as assessing trajectories in physics problems or real-world applications like ballistics.