A curved trajectory

A multiflash photograph can reveal details of the path, or trajectory, of a projectile. Figure 2.27 shows the trajectories of a projectile – a bouncing ball. Once the ball has left the child’s hand and is moving through the air, the only force acting on it is its weight.
The ball has been thrown at an angle to the horizontal. It speeds up as it falls – you can see that the images of the ball become further and further apart. At the same time, it moves steadily to the right. You can see this from the even spacing of the images across the picture.
The ball’s path has a mathematical shape known as a parabola. After it bounces, the ball is moving more slowly. It slows down, or decelerates, as it rises – the images get closer and closer together.
We interpret this picture as follows. The vertical motion of the ball is affected by the force of gravity, that is, its weight. When it rises it has a vertical deceleration of magnitude g, which slows it down, and when it falls it has an acceleration of g, which speeds it up. The ball’s horizontal motion is unaffected by gravity.
In the absence of air resistance, the ball has a constant velocity in the horizontal direction. We can treat the ball’s vertical and horizontal motions separately, because they are independent of one another.

Components of a vector

In order to understand how to treat the velocity in the vertical and horizontal directions separately we start by considering a constant velocity.
If an aeroplane has a constant velocity v at an angle $\theta $ as shown in Figure 2.28, then we say that this
velocity has two effects or components, ${v_N}$ in a northerly direction and ${v_E}$ in an easterly direction. These two components of velocity add up to make the actual velocity v.
This process of taking a velocity and determining its effect along another direction is known as resolving the velocity along a different direction. In effect, splitting the velocity into two components at right angles is the reverse of adding together two vectors – it is splitting one vector into two vectors along convenient directions.

To find the component of any vector (for example, displacement, velocity, acceleration) in a particular direction, we can use the following strategy:
Step 1: Find the angle $\theta $ between the vector and the direction of interest.
Step 2: Multiply the vector by the cosine of the angle $\theta $.
So the component of an object’s velocity v at angle $\theta $ to v is equal to $v\,\cos \theta $ (Figure 2.28).

Question

 

21) Find the x- and y-components of each of the vectors shown in Figure 2.29. (You will need to use a protractor to measure angles from the diagram.)

Figure 2.29: The vectors for Question 21