Problem 38 - Master Resource Book in JEE Main Physics

Question

A bullet emerges from a barrel of length \(1.2 \mathrm{~m}\) with a speed of \(640 \mathrm{~ms}^{-1}\). Assuming constant acceleration, the approximate time that it spends in the barrel after the gun is fired is [WB JEE 2008] (a) \(4 \mathrm{~ms}\) (b) \(40 \mathrm{~ms}\) (c) \(400 \mathrm{~ms}\) (d) \(1 \mathrm{~s}\)

Step-by-Step Solution

Answer

The approximate time is (a) 4 ms.

Step 1: Identify given information

The problem gives us the length of the barrel \( L = 1.2 \, \text{m}\) and the final speed of the bullet \( v = 640 \, \text{ms}^{-1} \). The initial speed \( u \) is \( 0 \, \text{ms}^{-1} \) as the bullet starts from rest.

Step 2: Choose the appropriate kinematic equation

We need to find the time \( t \) using constant acceleration. The kinematic equation that relates distance \( s \), initial velocity \( u \), final velocity \( v \), and time \( t \) is \( s = ut + \frac{1}{2} a t^2 \).

Step 3: Simplify with known initial conditions

Since the initial velocity \( u = 0 \), the equation simplifies to \( s = \frac{1}{2} a t^2 \). This gives us \( L = \frac{1}{2} a t^2 \) for \( t \), where \( s = L = 1.2 \, \text{m}\).

Step 4: Use the final velocity equation to find acceleration

The equation \( v = u + at \) with \( u = 0 \) simplifies to \( v = at \). Therefore, \( a = \frac{v}{t} = \frac{640}{t} \).

Step 5: Substitute \( a \) in the simplified distance equation

Substitute \( a = \frac{640}{t} \) in the distance equation \( L = \frac{1}{2} \cdot \frac{640}{t} \cdot t^2 \). This simplifies to \( 1.2 = \frac{640}{2t} \cdot t^2 \), which simplifies further to \( 1.2 = 320t \).

Step 6: Solve for time \( t \)

Solving \( 1.2 = 320t \) gives \( t = \frac{1.2}{320} \). Calculating this gives \( t = 0.00375 \, \text{s} \) or \( t = 3.75 \, \text{ms} \).

Step 7: Round and choose the closest option

Since \( 3.75 \, \text{ms} \) is closest to \( 4 \, \text{ms} \) in the options given, the answer is (a) \( 4 \text{ ms} \).

Key Concepts

Constant acceleration

When dealing with motion, constant acceleration is a critical concept that simplifies calculations. Constant acceleration means the velocity of an object changes at a steady rate over time. This is crucial in kinematic problems, like the bullet example, because it allows us to use specific equations to find unknown variables. For example, if you know the initial velocity, final velocity, and distance traveled, you can calculate the time of travel using the appropriate kinematic equations.

Constant acceleration can occur in various situations, such as a car speeding up uniformly or an apple falling vertically under gravity. Understanding the behavior of objects under constant acceleration is fundamental in physics and can help predict how objects move in the real world.

Key points about constant acceleration:

• The rate of change of velocity remains the same throughout the motion.
• Enables the use of kinematic equations to predict future motion or determine specific parameters.
• Commonly observed in free-fall situations and uniformly accelerating systems.

Kinematic equations

Kinematic equations are powerful tools in physics used to analyze motion when the acceleration is constant. These equations relate five primary variables: displacement ( s ), initial velocity ( u ), final velocity ( v ), acceleration ( a ), and time ( t ). When working with problems involving these variables, knowing at least three of them allows you to calculate the others.

In situations like the bullet emerging from a barrel, you often use the kinematic equation:

• \( s = ut + \frac{1}{2}at^2 \)
• \( v = u + at \)

This set of equations allows you to substitute known quantities to solve for unknowns effectively. For the bullet's problem, using these equations helped us determine the time the bullet spent in the barrel.

Tips for using kinematic equations:

• Ensure acceleration is constant - these equations do not work if acceleration varies.
• Identify what you know and what you need to find - this helps you choose the correct equation.
• Substitute known values and solve - keep units consistent throughout the calculations.

Projectile motion

Although not directly applied to the bullet problem, understanding projectile motion deepens comprehension of motion concepts. Projectile motion refers to the path taken when an object is thrown or launched into the air, moving under the influence of gravity alone. It is a type of motion experienced by any object launched into the air, ignoring other forces like air resistance.

A typical projectile motion scenario involves:

• An initial launch velocity that can have both horizontal and vertical components.
• Horizontal motion occurring at a constant velocity if air resistance is negligible.
• Vertical motion influenced by gravity, changing the object's velocity over time with constant acceleration down to earth.

By decomposing this motion into horizontal and vertical components, we can apply kinematic equations separately to each dimension.

Understanding projectile motion is essential for accurately predicting where an object will land and how long it will stay in the air, which is indispensable for fields ranging from sports to engineering.