## Sunday, October 23, 2016

### 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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