# A question of kinematics involving a changing velocity vector

In summary: To determine the positions at an arbitrary time. Think of displacements in very short time interval dt, and the component of the displacements in the direction of the line connecting A and B. You want to know how much the distance between A and B changes in this very short time interval.
Homework Statement
Point A moves uniformly with velocity v so that the vector v is continually "aimed" at point B which in its turn moves rectilinearly and uniformly with velocity u<v. At the initial moment of time v is perpendicular to u and the points are separated by a distance l. How soon will the points converge ?
Relevant Equations
distance = integral of velocity with time.
The statement "at the initial moment of time v ⊥ u and the points are separated by a distance l " gives us a picture like the one which I have added in attachment.
As the time passes velocity vector v would gradually change from fully vertical to fully horizontal in order to meet point B. Now, at any instant of time we have x-component of v as v cosθ and y-component as v sinθ. To catch point B our point A has to travel a distance of l in y- direction and to catch it in x-direction it has to cover the same distance as point B has covered. So, let's assume that they meet at time t , then

$$\int_{0}^{t} v cosθ dt$$ = ut

$$\int_{0}^{t} v sinθ dt$$ = l

Now, here I'm stuck. I know that θ is a function of time but I can't figure out which function is it. I know that x-component of v is gradually increasing sinusoidally and y-component is decaying sinusoidally but I'm not able to solve after that. I want to how can I solve for t after I have got those two equations.

#### Attachments

• Screen Shot 2019-07-23 at 10.00.20 AM.png
4.4 KB · Views: 307
• Screen Shot 2019-07-23 at 10.26.14 AM.png
9.5 KB · Views: 285
Think about how each velocity affects the distance apart.

haruspex said:
Think about how each velocity affects the distance apart.
Can you please elaborate just a little more?

Can you please elaborate just a little more?
In a short period of time, dt, by how much does the motion of B change the distance between them?

haruspex said:
In a short period of time, dt, by how much does the motion of B change the distance between them?
I’m sorry but
haruspex said:
In a short period of time, dt, by how much does the motion of B change the distance between them?
I’m sorry but I’m unable to use your clue.
If at any arbitrary time t we have velocities :- vcos$$\theta$$. , vsin$$\theta$$. and u.
So, distances are given by multiplying each instantaneous velocity by dt but I’m unable to keep track of l. How to write instantaneous distance in terms of l?
Thank you.

I’m sorry but

I’m sorry but I’m unable to use your clue.
If at any arbitrary time t we have velocities :- vcos$$\theta$$. , vsin$$\theta$$. and u.
So, distances are given by multiplying each instantaneous velocity by dt but I’m unable to keep track of l. How to write instantaneous distance in terms of l?
Thank you.
Don't work in terms of velocity components in the fixed XY frame. That just confuses things.
In time dt B moves distance udt to some location B'. What, roughly, is the angle ABB'?

haruspex said:
Don't work in terms of velocity components in the fixed XY frame. That just confuses things.
In time dt B moves distance udt to some location B'. What, roughly, is the angle ABB'?
Is angle ABB' going to be 90 degrees?

Is angle ABB' going to be 90 degrees?
Right. So how much does the distance udt that B moves affect its distance from A?

you need a good sketch.

Where ##\Delta l## represents the decrement in the distance between both points.
I added the ##-## sign before ##?_2## because it is decreasing the distance.
Do not get discouraged by the problems in Irodov's book, they're not easy.You'll nurture the intuition for these kind of problems where you need to seek out how the distance or whatever is changing. The problem before it (1.12) can be solved the same way, you need to find how the distance between two points is changing.

The model, which I have gained from your answer, is like this

but the problem is I'm not able to tell what A'B' would be? This is because at an arbitrary time I can't tell what AB is? Even if I use the information provided by haruspex that angle ABB' is roughly 90 degrees then also how can I solve for A'B'?
I want to thank you for giving me some attention, I'm really grateful to you. You know there are solutions present on internet (https://crackthequestion.blogspot.com/2008/06/q-113-in-general-physics-by-ie-irodov_20.html) but the problem is that they are using relative velocity concept (the component of u in the direction of v) just for the sake of solution, there is ground for doing it. What we should do is to take the component of v in the direction of u, but this approach is failing to give the answer. I really thank you for giving me your precious time, I earnestly request you to help through this question.

View attachment 247043

The model, which I have gained from your answer, is like this
but the problem is I'm not able to tell what A'B' would be?

Do not want to determine the positions at an arbitrary time. Think of displacements in very short time interval dt, and the component of the displacements in the direction of the line connecting A and B. You want to make that distance zero, that is why you need those components.
It is clear that motion of A diminishes the distance r between A and B by vdt. How much does the motion of B change the distance r in dt time? See the orange triangle.The velocity
vectors u and v enclose the angle theta, the same angle, v makes with the horizontal.

but the problem is I'm not able to tell what A'B' would be?
I was careful to ask, first, about the distance AB', not A'B'. And it will help if you draw the 90 degree angle more accurately.

## 1. What is kinematics?

Kinematics is the branch of physics that studies the motion of objects without considering the forces that cause the motion.

## 2. What is a velocity vector?

A velocity vector is a mathematical representation of an object's speed and direction of motion. It is usually represented graphically as an arrow pointing in the direction of motion with its length representing the speed.

## 3. How does the velocity vector change?

The velocity vector changes when there is a change in either the speed or direction of an object's motion. This change can be caused by external forces acting on the object, such as gravity or friction.

## 4. What is meant by a question of kinematics involving a changing velocity vector?

This refers to a problem or question that requires the use of kinematics principles to analyze the motion of an object with a velocity vector that is changing over time. It may involve finding the object's position, velocity, or acceleration at a specific point in time.

## 5. What are some real-life examples of kinematics involving a changing velocity vector?

Some examples include a car accelerating or decelerating on a highway, a ball being thrown in the air, or a rollercoaster going through loops and turns. Any situation where an object's velocity is changing can be analyzed using kinematics principles.

Replies
7
Views
499
Replies
10
Views
702
Replies
9
Views
998
Replies
2
Views
1K
Replies
10
Views
2K
Replies
23
Views
725
Replies
9
Views
6K
Replies
2
Views
2K
Replies
31
Views
646
Replies
1
Views
2K