Showing posts with label Relative velocity. Show all posts
Showing posts with label Relative velocity. Show all posts

Motion in One Dimension Problems with Solutions Eleven

We are solving problem in one dimensional motion. Here the body is moving along only one dimension. We do use four equations of motion. It is given in the problem that a string is connected over a pulley and force is applied on both of them with uniform velocity. A mass is connected with them which moves upward with the a constant velocity and we need to find out that velocity.

The force applied on the rope connected over the pulleys do pull the rope in downward direction so the arranged mass shall move in the direction.


Solution

Let the velocity of the block is V in upward direction. The velocity of the block along the horizontal direction can be found by resolving into components. With the same velocity the string is moving in the downward direction. We can solve the problem as shown in the diagram below.


Problem

It is given in the problem that the position vector varies with time as shown in the diagram below. It is given in terms of maximum position and time. We need to measure the distance covered during the time interval in which the particle returns to its initial position.


Solution
The position vector is given in terms of time and by differentiating with time, we can get velocity of the body. We need to know after how much time will come to rest. We shall equate the velocity to zero so that we can get the time in which the body velocity is zero and hence we can measure the distance in the mean time.

By substituting the time in the distance equation, we can measure the distance and the total distance covered is the double of it. The solution for the problem is as shown in the diagram below.


Problem

The instantaneous velocity of the particle moving in a plane and it is given in the vector format. It is given that the particle is started from the origin and we need to know the trajectory of the particle.


Solution

We can identify that the components of the velocity. To find the acceleration of each component by differentiating the velocity components. By simplifying the equation further as shown in the diagram below, we can find the expression for the displacement as shown in the diagram below. 


Problem

The particle is at a height from the ground and the velocity of the projectile is as shown in the diagram below. It is in vector format and it has horizontal and vertical components. We need to measure the angle of the projection.


Solution

The particle has both horizontal and vertical components and as the time progresses, the velocity components do change as the time progresses. Once we know the final components of the velocity and we can further find the angle of projection as shown in the diagram below.



Related Posts

Motion in One Dimension Problems with Solutions Two

Motion in One and Dimension Problems with Solutions Three

Motion in One and Dimension Problems with Solutions Four

Motion in One and Dimension Problems with Solutions Five

Motion in One Dimension Problems with Solutions Six

Motion in One Dimension Problems with Solutions Seven

Motion in One Dimension and Two Dimension Problems with Solutions Eight

Motion in One Dimension Problems with Solutions Nine

Motion in One Dimension Problems with Solutions Ten

Motion in One Dimension Problems with Solutions Twelve


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Motion in One Dimension Problems with Solutions Ten

We would like to solve a problem basing on relative velocity concept. There are two particles separated from a origin and the two positions are perpendicular to each other. They are moving with a known velocity. We need to know the nearest distance between them. The problem is as  shown in the diagram below.


Solution

Let us consider that A and B are the two bodies in the given problem and they are located perpendicular to the origin. Let A has travelled for one second and hence it covers three meter in that time. As it is given that it is initially at ten meter from the origin, after one second it is at the distance of seven meter.

Simultaneously the body B has moved a distance four meter from the origin. So we can find the shortest distance between the two bodies as shown in the diagram below.


Problem

In this one dimensional motion problem, the position of the body is given in terms of time and constants as shown in the diagram below.We need to know the particular value of the time at which acceleration of the body is zero.

Solution

It is given that displacement in terms of time.By differentiating displacement once with respect to time we can velocity and by differentiating velocity with respect to time we get acceleration.We shall equate acceleration to zero as per given problem to get the time required in the problem.

By using the basic rule of differentiation, we can solve the problem as shown in the diagram below.


Problem

This problem is also a similar problem to the above. We need to know what is the displacement of the particle at the instant when the velocity of the body is zero.


Solution

The given equation is in terms of displacement and by rearranging them we can get the displacement equation in terms of time as shown in the diagram below.

By differentiating the displacement once with respect to time we get velocity. As per the condition given in the problem, we need to equate it to zero. Thus it is found that velocity of the body becomes zero after three seconds. By substituting that value in the displacement equation, we can get the displacement as  shown in the diagram below.


Problem

This problem is also about the relation between displacement and time with some constants involved. The problem is as shown in the diagram below. We need to find the displacement after  a certain time to verify the first option.

We need to check the velocity to verify the second option. The third option is regarding the new new displacement of the particle.


Solution

By differentiating the displacement once with respect to time as shown in the diagram below, we can get the velocity of the particle. By differentiating once again, we can get acceleration of the same particle. By substituting the given condition of the time, we can solve the problem as shown in the diagram below.


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One Dimensional and Two Dimensional Motion Complete Lessons

A body moving only along one direction that is either along the x-axis are long y-axis is called as a one-dimensional motion.

If a body has to move along the horizontal direction, there may be need of some force that is acting along the direction of motion so that it is having some acceleration in the direction.

A body can also move along two directions, when two forces are acting on the body into different directions simultaneously and a typical example for this kind of motion is a projectile motion.

A physical quantity who has both magnitude and direction and which satisfies the rules of the vectors is called as vector.

Here are some of the posts that are linked with this concepts in this blog.


 Equations of Motion in One Dimension

Average Speed Average Velocity and Acceleration

Horizontal Projectile, Applications Problems with Solutions

Other Complete Lessons in this blog includes 


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Relative Velocity and Motion of a Boat across a River

Relative velocity is the comparative velocity of one body with respect to other body. One body will have a relative velocity with respect to other one only when there is a relative motion between them. If two bodies are having the motion in the same direction relative velocity of one with respect to other is the difference between them.It is simply because one bodies having more velocity when compared with the other body and how much more can be obtained only by subtracting that from the other value.

If two bodies are travelling in the opposite direction we can incorporate the same concept but being the bodies are in the opposite direction the resultant will become automatically the some of the two vectors as shown.

If two vectors are having an angle between them to find the relative velocity between them we shall incorporate a third body in between them. Generally the third body is the ground.

Expressing velocity of the any body with respect to ground is more appropriate than expressing the velocity of the ground with respect to other body.It is simply because it is not the ground that is moving but the body moving on the ground.



Problem and Solution

 Basing on a relative motion of two bodies. Let us consider two bodies each having the same velocity 10 m/s,one moving along the east other moving along the north from the same point. Find the relative velocity of one body with respect to other body?

While we are solving the problem we need to take a reference and the ground into consideration as reference.After identifying the answer,we can identify the direction of the vector.



Problem

The person is walking in the rain feel that the velocity of the rain is as twice as his velocity. It which angle you should hold his umbrella with vertical if he’s moving in a forward direction and training is happening in a vertical downward direction therefore he cannot be drenched in the rain?

Solution

This problem can also be solved basing on the concept of relative velocity as shown below. Whenever requirement of the third body is there we always get the ground into consideration because it is always there. We prefer to say the velocity of the body with respect to ground than in the  reverse way because bodies move on the ground.



Motion of a boat in a river

There are four different possible pace of about moving across the river.

Case one

 When a boat is crossing the river along the direction of the river:

In this case the motion of the boat is bit easy because it is supported by the stream of water therefore the boat takes less time to cross the river.

Case two

 Let us consider a case that the boat is moving against the steam of the river. In this case as boat has to overcome the opposition of the river it takes more time.



Case three

 Let us consider a case boat has to cross the river in such a way that it has to reach the exact opposite position.In this case we are actually not going along the river but where crossing the river.

 If you travel straight to the opposite point as the river pushes you are not going to reach the exact opposite point. That’s why we shall drive river boat with an angle  with the vertical.

 In the following derivation we have discussed the that with what angle he shall drive therefore he will be reaching the exact opposite point.In this case you are going to reach the exact opposite point means the path is the shortest but it is going to take the longest time to cross the river.


Case four

 In this case boat will go straight to the opposite point and being the river is going to push it it is not going to reach the exact opposite point but some other point in the bank of the river. In this case the party is going to be longest but the time is going to be shortest.

So here we have two choices.When you want to cross the river and the shortest path we have two choose case three where as when you want to cross the river with the shortest time we had to choose case four.




Case five

 Suppose you are crossing the river in such a way that you are making an angle  with the vertical, but it is not sufficient to reach the exact opposite position.

 In this case we have resolve the component of velocity of the boat along a horizontal and vertical parts. The vertical part will help you to identify the time taken to cross the river whereas the horizontal component of velocity has to be subtracted from the velocity of the river value while calculating the drift of the water. Drift is simply a particular value of the displacement because of which the boat has missed the exact opposite position. The equation for this drift is as shown.



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