Problem 22 - Master Resource Book in JEE Main Physics

Question

A police van moving on a highway with a speed of \(30 \mathrm{~km} / \mathrm{h}\) fires a bullet at a thief's car speeding away in the same direction with a speed of \(192 \mathrm{~km} / \mathrm{h}\). If the muzzle speed of the bullet is \(150 \mathrm{~m} / \mathrm{s}\), with what speed does the bullet hit the thief's car? (Note Obtain that speed which is relevant for damaging the thiefs car. [NCERT] (a) \(105 \mathrm{~m} / \mathrm{s}\) (b) \(100 \mathrm{~m} / \mathrm{s}\) (c) \(95 \mathrm{~m} / \mathrm{s}\) (d) \(110 \mathrm{~m} / \mathrm{s}\)

Step-by-Step Solution

Answer

The bullet hits the car with a speed of 105 m/s.

Step 1: Convert Speeds to Consistent Units

First, convert the speeds of both the police van and the thief's car from kilometers per hour into meters per second, since the muzzle speed of the bullet is given in meters per second. Use the conversion factor: \[1 \, \text{km/h} = \frac{1}{3.6} \, \text{m/s} \]- Police van speed: \[30 \, \text{km/h} \times \frac{1}{3.6} \approx 8.33 \, \text{m/s} \]- Thief's car speed: \[192 \, \text{km/h} \times \frac{1}{3.6} \approx 53.33 \, \text{m/s} \]

Step 2: Calculate Relative Speed

Calculate the speed at which the bullet approaches the thief's car by considering the relative motion between the bullet and the thief's car. Since both are moving in the same direction, take the difference in their speeds. The effective speed of the bullet relative to the police van is the muzzle speed. Thus, we have:- Bullet's speed relative to police van: \[150 \, \text{m/s}\]- Relative speed of the bullet to thief's car: \[(150 \, \text{m/s} + 8.33 \, \text{m/s}) - 53.33 \, \text{m/s} = 105 \, \text{m/s}\]

Step 3: Identify Correct Option

Now, match the calculated relative speed of the bullet with the thief's car to the given options in the question:The speed of the bullet relative to the thief's car is \(105 \, \text{m/s}\), so the correct answer is option (a) \(105 \, \text{m/s}\).

Key Concepts

Kinematics

Kinematics is a branch of physics that deals with the motion of objects. It focuses on understanding how objects move, ignoring the forces that cause this motion. The main components of kinematics include displacement, velocity, and acceleration.

Understanding these components is crucial for analyzing motion in everyday life as well as in various scientific fields. Displacement refers to the change in position of an object. Velocity, on the other hand, is the speed of an object in a particular direction. It is important in this scenario because both the police van and the thief's car are moving along the highway in the same direction.

• Displacement is the shortest path between two points.
• Velocity has both magnitude (speed) and direction.
• Acceleration is the rate at which velocity changes over time.

When calculating the velocity of objects moving in the same or opposite directions, understanding relative velocity, which compares the velocity of one object to another, is crucial. In this exercise, we calculate the speed at which the bullet approaches the thief's car by considering the relative velocities of the police van, the bullet, and the thief’s car.

Projectile Motion

Projectile motion refers to the movement of an object that is thrown or projected into the air, subject to only the force of gravity and air resistance. In this scenario, the bullet is in projectile motion once it is fired from the police van. The key aspect to understand here is that projectile motion involves two components: horizontal motion and vertical motion.

However, this particular problem focuses primarily on horizontal motion. The bullet maintains a constant horizontal velocity after being fired, altered only by its initial muzzle speed and the motion of the vehicles involved. This means, as the bullet travels towards the thief’s car, it does not change speed along the horizontal path unless external forces like friction or collision act on it.

• Horizontal motion involves constant velocity, assuming no air resistance.
• Vertical motion involves constant acceleration due to gravity.

Understanding projectile motion is essential for accurately predicting how fast the bullet will reach and interact with the moving car. It requires knowledge of relative motion to determine how the initial speed of the bullet and directional motion of both vehicles affect its travel path.

Units Conversion

Units conversion is a fundamental process in physics, allowing us to interpret speeds, distances, and other physical quantities in a consistent framework. In this problem, understanding how to convert from kilometers per hour (km/h) to meters per second (m/s) is critical because the bullet's speed is given in meters per second, necessitating unit consistency for accurate calculations.

• The conversion factor used is \(1 \, \text{km/h} = \frac{1}{3.6} \, \text{m/s}\).
• To convert from km/h to m/s, multiply by this factor.
This ensures that all speeds are in the same units, making mathematical operations, like subtraction necessary for relative speed calculations, straightforward. Consistency in units helps avoid confusion and errors, ensuring that when we determine the relative speed of the bullet to the thief's car, all parameters are harmonized and accurate. Proficiency in converting units is indispensable in physics and real-world problem-solving, ensuring results are precise and reliable.