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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