
A police party is chasing a dacoit in a jeep which is moving at a constant speed v. The dacoit is on a motorcycle. When he is at a distance x from the jeep, he accelerates from rest at a constant rate$\alpha $. Which of the following relations is true, if the police is able to catch the dacoit?
(a) ${v^2} \leqslant \alpha x$
(b ${v^2} \leqslant 2\alpha x$
(c) ${v^2} \geqslant 2\alpha x$
(d) ${v^2} \geqslant \alpha x$
Answer
126.6k+ views
Hint: The above problem can be solved by using the principle of kinematics. The police would catch the dacoit if the police cover the distance that is equal to the distance covered by the dacoit plus the initial distance between the car and jeep in the same duration.
Complete step by step answer
Given: The speed of the dacoit is v, the initial distance between the jeep and motorcycle is x, the acceleration of the motorcycle is $\alpha $.
The distance covered by the dacoit on the motorcycle is given as:
$d = vt......\left( 1 \right)$
The distance covered by the jeep to catch the dacoit is given as:
$X = \dfrac{1}{2}a{t^2} + x......\left( 2 \right)$
Equate the equation (1) and equation (2) to find the required relation.
$X = d$
$\dfrac{1}{2}\alpha {t^2} + x = vt$
$\alpha {t^2} + 2x = 2vt$
$\alpha {t^2} - 2vt + 2x = 0......\left( 3 \right)$
The police catch the dacoit if the roots of the quadratic equation (3) are real and unequal. The discriminant of the quadratic equation for real and unequal roots is given as:
$D \geqslant 0$
The discriminant of the quadratic equation (3) is given as:
${\left( { - 2v} \right)^2} - 4\left( \alpha \right)\left( {2x} \right) \geqslant 0$
${v^2} - 2\alpha x \geqslant 0$
${v^2} \geqslant 2\alpha x$
Thus, the true relation for catching the dacoit is ${v^2} \geqslant 2\alpha x$ and the option (c) is the correct answer.
Note: The above problem can also be solved by using the concept of the relative motion. The dacoit can be assumed stationary at some separation and police moves relative to the dacoit.
Complete step by step answer
Given: The speed of the dacoit is v, the initial distance between the jeep and motorcycle is x, the acceleration of the motorcycle is $\alpha $.
The distance covered by the dacoit on the motorcycle is given as:
$d = vt......\left( 1 \right)$
The distance covered by the jeep to catch the dacoit is given as:
$X = \dfrac{1}{2}a{t^2} + x......\left( 2 \right)$
Equate the equation (1) and equation (2) to find the required relation.
$X = d$
$\dfrac{1}{2}\alpha {t^2} + x = vt$
$\alpha {t^2} + 2x = 2vt$
$\alpha {t^2} - 2vt + 2x = 0......\left( 3 \right)$
The police catch the dacoit if the roots of the quadratic equation (3) are real and unequal. The discriminant of the quadratic equation for real and unequal roots is given as:
$D \geqslant 0$
The discriminant of the quadratic equation (3) is given as:
${\left( { - 2v} \right)^2} - 4\left( \alpha \right)\left( {2x} \right) \geqslant 0$
${v^2} - 2\alpha x \geqslant 0$
${v^2} \geqslant 2\alpha x$
Thus, the true relation for catching the dacoit is ${v^2} \geqslant 2\alpha x$ and the option (c) is the correct answer.
Note: The above problem can also be solved by using the concept of the relative motion. The dacoit can be assumed stationary at some separation and police moves relative to the dacoit.
Recently Updated Pages
JEE Main 2023 (April 8th Shift 2) Physics Question Paper with Answer Key

JEE Main 2023 (January 30th Shift 2) Maths Question Paper with Answer Key

JEE Main 2022 (July 25th Shift 2) Physics Question Paper with Answer Key

Classification of Elements and Periodicity in Properties Chapter For JEE Main Chemistry

JEE Main 2023 (January 25th Shift 1) Maths Question Paper with Answer Key

JEE Main 2023 (January 24th Shift 2) Chemistry Question Paper with Answer Key

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility & More

JEE Main Login 2045: Step-by-Step Instructions and Details

Class 11 JEE Main Physics Mock Test 2025

JEE Main Exam Marking Scheme: Detailed Breakdown of Marks and Negative Marking

JEE Main 2023 January 24 Shift 2 Question Paper with Answer Keys & Solutions

JEE Mains 2025 Correction Window Date (Out) – Check Procedure and Fees Here!

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced 2025: Dates, Registration, Syllabus, Eligibility Criteria and More

NCERT Solutions for Class 11 Physics Chapter 1 Units and Measurements

NCERT Solutions for Class 11 Physics Chapter 9 Mechanical Properties of Fluids

Units and Measurements Class 11 Notes: CBSE Physics Chapter 1

NCERT Solutions for Class 11 Physics Chapter 2 Motion In A Straight Line
