## Saturday, October 22, 2016

### Motion in One Dimension and Two Dimension Problems with Solutions Eight

We would like to solve the problem in two dimension. In the given problem, it is given in that a stone is projected with an angle 45 degree from the horizontal and we need to know the angular momentum of the body when it has reached its maximum height.  The problem is as shown in the diagram below.

Solution

We know that the projectile is a body moving along two dimensions and we know that the range is the maximum horizontal distance that it travels and time of flight is the time for which the body is in air. We know that when the projectile reaches its maximum height, there it has only horizontal velocity and hence it is the cos component of the velocity. We also know that the body is at the maximum height and we know the equation of that for a projectile. Angular momentum is defined as the moment of momentum. It is product of momentum and the distance where we need to measure the angular momentum from a reference point. Thus by substituting the values, we can get angular momentum value as shown in the diagram below.

Problem

This problem is regarding horizontal projection. It has only initial horizontal velocity and it thrown from a certain known height. If the line of joining of the point of projection to the point where it hits the ground makes an angle 45 degree, we need to find the initial velocity of the projection. The problem is as shown in the diagram below.

Solution

The body is initially has only initial velocity. As the time progresses, the projectile will be start getting velocity  along y direction also due to acceleration due to gravity.

After a certain time, the body has both horizontal and vertical velocity along two perpendicular directions. So to measure the effective velocity, we need to use the vector method and we can find the resultant velocity as shown in the diagram below.

As the angle is 45 degree of landing, the vertical and horizontal components of the velocities are same and we can use that concept and solve the problem as shown in the diagram below.

Problem

It is given in the problem that the three particles stats from the origin. The first one is moving along x axis, the second one  is along y axis and the third one is moving in the plane of both of them.We need to know what is the velocity of the third particle so that these three particles lie in the same line. The problem is as shown in the diagram below.

Solution

We can represent the velocity of the first body along X axis, velocity of the second body along Y axis and the third one is moving along the plane as shown in the diagram below. At a time t, the positions of the three bodies can be represent as shown in the diagram. As they have to stay in the same line, the slope of the graph at both the points has to be the same.

We can further solve the problem as shown in the diagram below.

Problem

It is given in the problem that an aircraft is flying at uniform speed. If it makes same angle at two different points on the ground. We need to know the height of the aircraft at that instant. The problem is as shown in the diagram below.

Solution

Let the total angular separation between then is angle and hence the angle with the vertical is half of it. We can  define the angle and write the equation as shown. We can find the horizontal distance as the product of speed and time. We can further simplify the problem as shown in the diagram below.

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