## Tuesday, October 25, 2016

### Motion in One Dimension Problems with Solutions Twelve

We are interested in solving the problems based on displacement,velocity and acceleration. They are all belongs to one dimensional motion. We know that the velocity is the rate of change of displacement and acceleration is the rage of change of velocity.

In the first problem it is given that position of the particle is given in the XY plane and it is given in terms of time and trigonometric function. We need to know the path of the particle and the problem is as shown below.

Solution

Position vector is given to us in terms of both X and Y components. By multiplying with I unit vector we identify it as the X component. Similarly by multiplying with J, we can identify the component as Y component. J is a unit vector along Y axis. Unit vector is a vector with specific direction and has magnitude of only one unit.

Using trigonometric rule, we can find the relation between X and Y as shown in the diagram below. The path is a circle as the equation supports that mathematically.

Problem

A particle moves according to the equation given in the problem with some constants. We need to find the velocity of the particle in terms of time and we can assume that the body starts from rest. The problem is as shown in the diagram below.

Problem

Rate of change of velocity is given in the problem and we can rearrange the terms to get the component of velocity. But we need the total velocity and to get that we shall integrate the component of velocity with certain given limits. By simplifying the equation further, we can solve the problem as shown in the diagram below.

Problem

A particle moves in a straight line and the position of the particle is given to us in terms of time as shown in the diagram below. We need to know how the velocity and acceleration of the particle changes with respect to time and we need to check with the options given in the diagram below.

Solution

What is given in the problem is in terms of displacement and to get the velocity we need to differentiate the equation. It can be found further that after two seconds velocity of the particle and its displacement becomes zero.

By differentiating the velocity further we can get acceleration as shown below. It can be concluded basing on the equations that if time is less than three seconds, acceleration could be negative.

Problem

It is given in the problem that after the engine is switched off, the boat goes for retardation and it is given as shown in the diagram below. We need to know the velocity of the particle after a specified time.

Solution

By rearranging the terms, we can get a part of velocity. To get the total velocity in the given limits, we need to integrate the equation. Thus we get the velocity of the particle at a given time as shown in the diagram below.

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