Let the body play as one
The tennis ball travels as an expression of your motion.
Any person picking up the racquet for the first time will intuitively move their arm so that the racquet meets the ball. The racquet moves to intercept the tennis ball, and the ball responds by going in the direction of the swing path. The natural thing we do is that we swing where we want the ball to go. For players just starting tennis, it might be a general direction. Precision comes with greater awareness after enough experience playing.
If you simply throw the ball, you will move your arm in the direction you want the ball to travel. If you're pushing a shopping cart, you're doing so by moving your body where you want the cart to go. By extension of your motion, everything you connect with responds in principle as described by laws of physics.
Another way to look at it, is that what you put in, you get some form of an output. This is natural and built into our intuition, and if we let ourselves feel and explore our motion in relation to other objects in the flow of now, then we develop a strong awareness of how things work in this world.
Now imagine you want to throw the ball at the furthest point reachable, say across five tennis courts. You will most likely use the rest of your body and not just your arm. You might even start shuffling in the same direction before accelerating your arm as fast as you possibly can. You will intuitively begin from a sideways position so you can add rotation, which supports greater acceleration for your arm. Your entire body participates to make happen what your mind is looking to do.
Playing tennis is no different. The motion you create is an influence on the ball's motion. You can use your entire body's motion or part of it. At the very minimum, you will swing your arm and make contact with the ball to send it over the other side of the court. Now, if the ball only receives input from your arm, then the output will be expressed by what your arm can do on its own. When we push something heavy across the room, we get behind the object and engage our entire body. We start with generating force using the ground. Legs pushing against the ground create a reaction of equal measure in the opposite direction. As a result, our body moves in the direction we want the object to go.
Generating motion from the ground up with your legs is also the most efficient way of getting everything else moving. It creates a singular direction for your entire body, so what you're hitting will respond with clarity and purpose. The arms travel together with the rest of the body.
So, to get the most out of your tennis game, you should always play by letting your body work together as one. Notice that professional tennis players set themselves up to engage their legs and move the entire body as one in the direction where they want the tennis ball to go. All the work they do with their footwork, timing their preparation, is to make that execution happen so that their body can participate as one to make the most of the shot.
If you're a tennis player, then simply be aware of involving your legs in every shot. Explore this and you will come to understand how this works. Eventually, involving your entire body for each shot will become intuitive. You will also begin to organize yourself better between your shots to get behind the ball each time.
If you're a tennis coach, you can help your students by directing their attention to this consistently. They will improve at a much higher rate because this is the essence of the game, generating motion to get the ball to go where intended.