JEE Main 2020, 8 Jan Shift-I
LEVELJEE Main
Visualized Solution
The Sigma Insight: Relative Velocity
The Dance of Two Particles
Imagine you are standing at the origin of a massive coordinate system. You see two particles, let's call them Particle A and Particle B, moving along the axes. Particle A is constrained to the X-axis, performing a dance dictated by the equation . Meanwhile, Particle B is sliding along the Y-axis, following its own rhythm given by .
The question asks us for a very specific perspective: what is the speed of Particle B if you were sitting on Particle A at the exact moment ? This is a classic relative motion problem, and to solve it, we first need to figure out how fast each particle is moving individually.
Unlocking the Velocities
In kinematics, position is just the starting point. To find the velocity, we need to look at the rate of change of position. Mathematically, this means taking the derivative of the position equations with respect to time.
Let's start with Particle A. Its position is . Differentiating this gives us its velocity:
Since Particle A is moving along the X-axis, its velocity vector is .
Now, let's look at Particle B. Its position is . Differentiating this gives us:
Since Particle B is moving along the Y-axis, its velocity vector is .
Freezing Time at
The problem asks for the relative speed at a specific instant: . Let's plug this time into our velocity equations to see exactly what's happening at that moment.
For Particle A:
So, . Particle A is moving to the right at a leisurely pace.
For Particle B:
So, . Particle B is zooming downwards along the Y-axis!
The Relative Perspective
Now comes the crucial step. We need the velocity of Particle B as measured in the frame of Particle A. This is the relative velocity, denoted as . The formula for relative velocity is beautifully simple:
Substituting our vectors into this equation:
This vector tells us that if you were sitting on Particle A, you would see Particle B moving to the left at and downwards at .
Calculating the Final Speed
The question doesn't just ask for the velocity vector; it asks for the speed, which is the magnitude of the velocity vector. To find the magnitude of , we use the Pythagorean theorem:
The problem states that this speed is equal to . By comparing our result with the given expression, we can easily see that:
Therefore, the value of is exactly 580.
This problem beautifully illustrates how calculus and vector algebra come together to solve relative motion scenarios. Always remember to find the individual velocity vectors first before jumping into the relative frame!
Similar Questions
Pathfinder for Olympiad and JEE Advanced Physics
LEVELJEE Advanced
Comprehension Passage
Two particle A and B are moving towards each other on a straight line with equal speeds . At an instant that is assumed , distance between the particles is . It is desired to move another particle C always maintaining a distance from the particle A and from the particle B.
Question 1:
When and for how long can the particle C fulfil the given condition?
* Multiple Correct Options
(A)
(B)
(C)
(D)
Question 2:
What is speed of the particle C at the instant ?
* Multiple Correct Options
(A)
(B)
(C)
(D)
Question 3:
What is modulus of acceleration of the particle C at the instant ?
* Multiple Correct Options
(A)
(B)
(C)
(D)
Question 4:
At the instant, when the line joining locations of A and B is perpendicular to the line joining locations of B and C, what are the magnitudes of velocities of C relative to A and B respectively?
* Multiple Correct Options
(A)
and
(B)
and
(C)
and
(D)
and
Pathfinder for Olympiad and JEE Advanced Physics
LEVELJEE Advanced
Two material particles A and B are moving in free space. How their position coordinates x, y and z vary with time t is shown in the following graphs. Determine at what instant of time the particles are closest to each other and the closest separation.
Pathfinder for Olympiad and JEE Advanced Physics
LEVELJEE Advanced
At the initial instant, two particles are observed at different locations moving towards each other with velocities and . If they are subjected to constant accelerations and in directions opposite to their initial velocities, they will meet twice. If time interval between these two meetings is , find suitable expression for their initial separation.
Pathfinder for Olympiad and JEE Advanced Physics
LEVELOlympiad
A particle P is moving with a constant speed on a straight line that makes an angle with the positive -direction of a coordinate system. When P crosses the -axis at a point , another particle Q starts from the origin and chases P with a uniform speed (). The chaser Q always maintains its velocity vector towards the chased P. (a) How long after Q starts from the origin, will it catch P? (b) If both the chaser Q and the chased P move with equal speeds (i.e. ), what will be the minimum distance between them and what will be the maximum magnitude of acceleration of the chaser Q?
JEE Advanced 2014
LEVELJEE Advanced
Airplanes and are flying with constant velocity in the same vertical plane at angles and with respect to the horizontal respectively as shown in figure. The speed of is . At time , an observer in finds at a distance of . This observer sees moving with a constant velocity perpendicular to the line of motion of . If at , just escapes being hit by , in seconds is
JEE Advanced 2003
LEVELJEE Advanced
A particle of mass , moving in a circular path of radius with a constant speed is located at point at time and a man starts moving with a velocity along the positive Y-axis from origin at time . Calculate the linear momentum of the particle w.r.t. man as a function of time.
Pathfinder for Olympiad and JEE Advanced Physics
LEVELJEE Advanced
Two cars are moving at constant speeds; one on a circular path of radius and the other on a straight road. Magnitude of velocity of one car relative to the other has been recorded at regular intervals of time and data thus obtained is represented in a graph as shown in the figure. Calculate speeds of both the cars relative to the ground.
JEE Main 2020, 2 Sep Shift-I
LEVELJEE Main
Trains and are running on parallel tracks in the opposite directions with speeds of and , respectively. A person is walking in train in the opposite direction to its motion with a speed of . Speed (in ) of this person as observed from train will be close to (Take, the distance between the tracks as negligible)
(A)
28.5
(B)
30.5
(C)
29.5
(D)
31.5
JEE Advanced 2022
LEVELJEE Advanced
List I describes four systems, each with two particles A and B in relative motion as shown in figure. List II gives possible magnitudes of their relative velocities (in ) at time .
JEE Advanced 2002
LEVELJEE Main