Problem 54 - Master Resource Book in JEE Main Physics

Question

A car is moving with a speed of \(100 \mathrm{kmh}^{-1}\). If the mass of the car is \(950 \mathrm{~kg}\), then its kinetic energy is (a) \(0.367 \mathrm{M}\rfloor\) (b) \(3.67 \mathrm{~J}\) (c) \(3.67 \mathrm{M} \mathrm{J}\) (d) \(367 \mathrm{~J}\)

Step-by-Step Solution

Answer

The answer is (a) \(0.367 \text{ MJ}\).

Step 1: Convert Speed to Meters per Second

The speed is given in kilometers per hour. We first need to convert this to meters per second to use it in the kinetic energy formula. Use the conversion factor: \(1 \text{ km/h} = \frac{1000}{3600} \text{ m/s}\). Thus, \(100 \text{ km/h} = 100 \times \frac{1000}{3600} \text{ m/s} = \frac{250}{9} \text{ m/s}\).

Step 2: Write the Formula for Kinetic Energy

The formula for kinetic energy \(KE\) is \(KE = \frac{1}{2}mv^2\), where \(m\) is mass and \(v\) is the speed. Here, \(m = 950 \text{ kg}\) and \(v = \frac{250}{9} \text{ m/s}\).

Step 3: Substitute the Values into the Formula

Substitute the given values into the kinetic energy formula: \[ KE = \frac{1}{2} \times 950 \times \left( \frac{250}{9} \right)^2 \]

Step 4: Simplify the Calculation

First, find \(v^2\): \[ \left( \frac{250}{9} \right)^2 = \frac{62500}{81} \] Now multiply by \(\frac{m}{2}\): \[ KE = \frac{1}{2} \times 950 \times \frac{62500}{81} = \frac{950 \times 62500}{2 \times 81} \]

Step 5: Calculate the Kinetic Energy

Calculate the numerical value: \[ KE = \frac{950 \times 62500}{162} \]First multiply and then divide: \[ 950 \times 62500 = 59375000 \] \[ KE = \frac{59375000}{162} \approx 366975 \]So, \(KE = 366975 \text{ J} = 0.367 \times 10^6 \text{ J} = 0.367 \text{ MJ}\).

Step 6: Choose the Correct Answer

Compare the calculated kinetic energy value with the options provided:(a) \(0.367 \text{ MJ}\)(b) \(3.67 \text{ J}\)(c) \(3.67 \text{ MJ}\)(d) \(367 \text{ J}\)The calculated kinetic energy \(0.367 \text{ MJ}\) corresponds to option (a).

Key Concepts

Conversion of Units

When solving problems in physics, it's crucial to ensure that units are consistent. In this exercise, the car's speed is given in kilometers per hour and needs to be converted to meters per second. This is essential because the kinetic energy formula requires speed in meters per second. To convert from - **kilometers per hour (km/h) to meters per second (m/s)**, use the conversion factor: \1 \text{ km/h} = \frac{1000}{3600} \text{ m/s}.\To illustrate:- For a speed of 100 km/h, apply the conversion: \( 100 \times \frac{1000}{3600} \text{ m/s} = \frac{250}{9} \text{ m/s}. \) It’s always good to know some basic conversion factors, as they frequently appear in physics problems.Always verify the units you end up with after conversion. This helps in accurately applying formulas and obtaining correct results.

Kinetic Energy Formula

Kinetic energy is a measure of the energy an object has due to its motion. The kinetic energy formula is derived based on mass and velocity, two components indicating how much force is involved in the motion.The formula for kinetic energy (KE) is:\[ KE = \frac{1}{2}mv^2 \] where:- \( m \) stands for the mass of the object and \( v \) represents its velocity.In the formula, the term \( \frac{1}{2} \) indicates that kinetic energy depends on half of the product of the mass and the square of the velocity. This underscores how slight changes in velocity significantly impact kinetic energy, as energy increases with the square of speed.For example, in this problem, with a car of mass 950 kg moving at \( \frac{250}{9} \) m/s, the kinetic energy is calculated as:\[ KE = \frac{1}{2} \times 950 \times \left( \frac{250}{9} \right)^2 \]This demonstrates how mass and velocity contribute to determining the energy available due to motion.

Motion in One Dimension

Understanding motion in one dimension simplifies the dynamics of an object's movement. In physics, this refers to movement along a single straight line, either in a horizontal or vertical direction. It involves variables such as speed, velocity, and acceleration confined to that line. For instance: - When analyzing the problem of a car moving with a specific speed, we treat it as a case of motion in one dimension. The car isn't changing direction or height, just moving straightly, which simplifies calculations. Different aspects of motion, like speed and acceleration, only need to consider their forward or backward direction along this one line. In summary, focusing on motion in one dimension reduces complexity and helps us analyze essential aspects of movement without extra dimensions complicating the calculations. This is vital when dealing with forces and energy along a linear path.