I am trying to do this textbook question at the end of a chapter on vectors:
A bomb is moving vertically downwards with a speed of 400 m/s. At a height of 20 m above the ground it explodes and all the particles are thereby given an additional speed 200 metres per second in all directions. Find the diameter of the smallest circle within which the particles will strike the ground (ignore the effect of gravity)
The book says the answer is 23.09 metres. I cannot see this.
Surely if the bomb explodes at 20 metres some particles will travel horizontally at 200 m/sec and vertically at 600 m/sec, reaching the ground after 1/30 of a second. in that time will they will cover a horizontal distance of 200/30 = 6.66 m so that all particles will strike the ground within a circle of diameter 13.32 m.
I cannot see where vectors come in.
The center of mass of the bomb has a downward velocity of $400$ before the explosion.
The center of mass of the cloud of particles still has a downward velocity of $400$ after the explosion. What is different is that each particle has a velocity of $200$ (in some direction) relative to the center of mass.
So there is a particle that is now traveling straight down at a speed of $400 - 200 = 200$ (velocity $200$ upward relative to the center of mass) and one traveling straight down at a speed of $400 + 200 = 600$. The others are all traveling with velocities that are the vector sums of a vector of size $400$ straight down and some vector of size $200$ in some other direction.
None of the particles has a horizontal velocity of $400.$ The greatest possible horizontal velocity is $200,$ and those are not the particles that strike the ground farthest from the center.
I hope by now you have noticed from comments and other answers that you do not have to think about "vectors" very much in order to solve this problem; you can just think about the vertical and horizontal components of motion and solve the problem algebraically.
But I think it can be useful to visualize this problem with vectors.
We start with a $400$ m/s vector straight down, representing the velocity of the center of mass. We add to this a $200$ m/s vector in an arbitrary direction. Some people like to put vectors tail-to-tail and draw a parallelogram to represent the vector sum, but in this case I think it helps to put the vectors head-to-tail, with the tail of the $200$ m/s vector at the head of the $400$ m/s vector as shown in the diagram below.
The third side of the triangle, the red vector in the diagram, is the vector sum. This vector gives the actual direction of travel of one of the particles.
In the diagram above, I have actually superimposed a scaled-down copy of this vector diagram onto a picture that includes the point where the bomb bursts above the horizontal ground below, with the tail of the blue vector at the location where the bomb bursts. The red vector then gives us the line along which that particular particle will travel (if there were no effect of gravity, as the problem tells us we should assume). You can see graphically where the particle then hits the ground.
Every other particle has some other $200$ m/s vector relative to the center of mass, but every such vector can be placed so that its tail is at the tip of the $400$ m/s vector and its tip is somewhere on the circle in the diagram. The path of every one of those particles, like the one shown, is a straight line starting at the tail of the $400$ m/s vector and passing through some point on the circle (whichever point the tip of that particle's $200$ m/s vector happens to land on).
The shower of particles therefore forms a cone. You can deduce the shape of that cone from the diagram, in fact you can easily compute the angle of the cone given the known velocities from the problem statement and a little trigonometry.
It is then easy to compute the maximum distance from the center of the circle on the ground to the point where any particle hits.