## Wednesday, October 26, 2016

### 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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