Showing posts with label Velocity. Show all posts
Showing posts with label Velocity. Show all posts

Equations of Motion along straight line and Freely Falling Video lesson

Equations of motion in one dimensional motion

We know that a body in one dimensional motion has displacement, velocity which is different at initial level and final level and hence the body is also having some acceleration. The body is covering some displacement in a specified time. Taking this into consideration, we can obtain the relation between the above mentioned physical quantities using the equations of motion. They relate some of the above mentioned physical quantities and the relation is among the four quantities. If we know any of the three, by using appropriate equation, we can find the unknown physical quantity using the relations available to us.




 Equation of motion for a freely falling body

A body starting from the state of rest from a certain height and falling vertically downwards under the influence of gravitational force is called as a freely falling body. For a freely falling body, initial velocity is zero and the displacement of the body is nothing but the height of fall in a given time. As the body falls down, its velocity increases and hence it is under acceleration and it is due to gravitational force. This acceleration due to the earth is called acceleration due to gravity and it is constant at a given place. Taking this into consideration, we can rewrite the equations of motion as shown in the video lecture below. The time taken by the body to reach the ground from the maximum height is called time of descent.



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Motion in One and two Dimensions Problems with Solutions Fourteen

We are solving problems on one and two dimensional motion in this series of chapters. Here we are dealing with motion of a body moving along only one dimension and also we are studying the motion of a body in a plane where it has motion both along X and Y axis simultaneously.

In the given problem a body is dropped from a certain height and it strikes the inclined plane at a height h above the ground. As a result of impact, velocity of the body becomes horizontal. We need to measure  that it will take maximum time to reach the ground basing on what condition. The problem is  as shown in the diagram below.


Solution

It is given that the body is dropped from a certain height and hence its initial velocity is zero. It has fallen only some distance before it actually strikes the ground. That can be found as the difference between the total height from where it  is dropped and the location where it hit the ground.

Thus we can measure the time of fall in both the cases using the equations of motion. We need to measure the total maximum journey time. For the parameter to be maximum, its differentiation function with height has to become zero as a mathematical rule. 

By simplifying it, we can find that relation as shown in the diagram below.


Problem

It is given in the problem that a projectile is projected with a velocity and a known angle of projection. It is given that the trajectory grazes the vertices of the triangle as shown in the diagram. We need to know the relation between the angles with the horizontal.



Solution

The diagram for the problem data is as shown in the figure. We can express the angles with the horizontal in terms of horizontal and vertical displacement as shown in the diagram below.

We also know that the vertical displacement can be expressed in terms of horizontal displacement and angle of projection in terms of range as shown further.

By comparing this two equations, we can find the relation as shown below.


Problem

Two cannons are installed at the top of a cliff of 10 meter and they fire a shot with a known speed at some interval of time. The first cannon is fired with an angle and the other one is done horizontally. We need to know where do the two cannons collide with each other. The problem is as shown in the diagram below.


Solution

Both of them were fired from a certain height. But one horizontally and the other with an angle. By substituting the given data, we can simplify the equations as shown in the diagram below.



Problem

It is given in the problem that a particle is projected from the ground with a initial speed at angle of projection with the horizontal. We need to measure the average velocity of the projection between its point of projection and the highest point of the trajectory. The problem is as shown in the diagram below.


Solution

We know that the average velocity is the ratio of total displacement of a body in the total time of the journey. We can find the total displacement of the journey as a vector sum of horizontal and vertial displacements and the time of journy is half of time of flight.By substituting that data, we can  solve the problem as shown in the diagram below.



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Motion in One Dimension Problems with Solutions Thirteen

We are solving series of problems in one dimensional motion. It is given in the problem that a point moves in a straightline under a retardation as given in the problem where k is a constant.Acceleration is defined as rate of change of velocity. If a body has acceleration means, its velocity is increasing with time. If it is deceasing with  time, it is called retardation.

It is given that the particle has initial velocity. We need to know the distance covered by it in the given time. The problem is as shown in the diagram below.


Solution

We know that retardation is a negative acceleration. Here in this case the velocity of the body is decreasing with time. To indicate it, a negative sign is taken into consideration. We can rearrange the velocity  in terms of time as shown in the diagram below. As we know the part of velocity, to get the total velocity by integrating it in the given limits. By using rules of integration, we can the value of final velocity after applying the limits of lower and upper.

We need displacement and hence write the velocity as the rate of change of displacement. We can write the equation for small displacement and by integrating it, we can get the displacement of the particle as shown in the diagram below.


Problem

In the arrangement shown in the figure and instantaneous velocities of two masses, we need to know the angle between the vertical as shown in the diagram below. 


Solution

Let the vertical distance between pulley and the surface is X. Let the horizontal distance is 2a. It is given that the length of the string is constant and hence it can be found in the given terms as shown in the diagram below. As the equation is constant, its differentiation with time is zero.

By applying that condition, we can the relation between two velocities as shown. Basing on that we can find the angle and the detailed solutions is given below.


Problem

It is given in the problem that a ball is thrown vertically upward with a known velocity. While going upward and again while coming back after reaching its maximum height, the ball came back to the same point and the time interval between them is given to us in the problem. We need to measure the initial velocity of the ball. The problem is as shown in the diagram below.


Solution

Let the height that is crossed twice is and the intervals are given in the problem as shown in the diagram. Thus we can equate the height equation both in terms of the first time and the second time. By solving these two  equations, we can solve the problem as shown in the diagram below.


Problem

It is given in the problem that a body is projected vertically and different points are given in the journey. If the body is released from position A and we need to know the time of descents ratio need to be found.


Solution

We know that the time intervals for each part is same and hence to reach each point, the time taken in multiples of that interval as shown in the diagram below. We need to use the second equation of motion and solve the problem as shown in the diagram below.


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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.



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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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Motion in One Dimension Problems with Solutions Nine

One dimensional motion is the study of motion of a body along only one dimension either along X axis or Y axis. We primarily use equations of motion  and they are four equations of motion. The problem deals with a body in the state of rest. It accelerates for some time, moves with a constant velocity for some more time and finally decelerates for some more time before coming to the state of the rest. Total time of the journey is given to us and average velocity of the body is also given to us. We need to know the time for which the body moves with constant velocity. The problem is as shown in the diagram below.


 Solution

There are three parts of motion in the problem. The first part is accelerated part, the second part is uniform velocity part and the third part is retardation part. We need to know the time interval for which the body is having uniform speed. If we assume that the uniform motion is happened for a given time t, then, we can find the time for accelerated time as shown in the diagram below.

We can find the final velocity of the body after acceleration and we can find the total distance travelled as the product of the total time of the journey and the average velocity. By solving the equation further, we can solve and find the time for which the body is in uniform velocity. The solution is as shown in the diagram below.


Problem

It is given in the problem that the body half of the distance has some velocity and remaining half of the distance can cover half of the part with one velocity and the second part with a different velocity. We need to find the average velocity of the system ?


Solution

For the first half of the distance, it moves with constant speed and hence  we can use the formula that the distance is the product of velocity and time. During the second half of the journey, the average velocity as the arthematic mean of the two velocities. Thus we can find the second half of the journey time. Thus substituting the data further we can  find the average velocity as shown in the diagram below.


Problem

A money is on the ground and it wish to climb to the top of a vertical pole of known height. It has a tendency of climbing 5 meter up and then one meter down in the same interval of time. If it is a continuous process, we need to know that how much time that the money is going to take to reach the top of the pole ?


Solution

The monkey has to cover 13 meter but in the last five seconds, it covers five meter. So it further need to cover a remaining distance of eight meter and the slip effect has to be considered only during this part. In the last phase of five meter, as it has all ready reached the top, we need not worry of slipping down. The solution is as shown in the diagram below.


Problem

It is given in the problem that a train is accelerates between two stations with acceleration, retardation and uniform velocity and the ratio of times for that is given to us as shown in the diagram below. The maximum speed of the body is given to us and we need to measure the average speed of the body in the given problem.



Solution

Using the data of the problem that the body has acceleration for one second, we can find the acceleration of the system. Thus we can find the average velocity as shown in the diagram below.


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