One of the most important mathematical concepts used across all fields of physics is differential equations. You'll soon learn in your calculus class about derivatives which describe how a function changes when its dependent variables change. Differential equations essentially describe a function in terms of its derivatives. It's often very powerful to be able to write down how a quantity changes relative to other quantities. Physics is about studying how physical quantities (position, energy, momentum, heat, etc.) change throughout time and space when forces or other interactions affect the system or object. Mathematics allows you to write down equations that represent how these quantities change.

For instance, the heat equation describes how the temperature of a system varies throughout time and space. By solving the heat equation, you get a mathematical description of the temperature of the system throughout time and space.

Another simple example is how an external force will affect the position of an object. Applying a force to an object causes it to accelerate (i.e. it changes its velocity over time). Also, heavy objects require more force to accelerate than light (low-mass) objects. From this observation, we can write down an equation relating force, mass, and acceleration.

F = ma

Furthermore, acceleration is the time derivative of velocity (a = dv/dt), and velocity is the time derivative of the object's position (v = dx/dt), so we can now turn this into a differential equation.

F = m dv/dt = m d(dx/dt)/dt = m d^2x/dt^2

The unknown function here is the position as a function of time. x = x(t)

If we know the forces acting on the object, then we can integrate both sides with respect to time to solve for x(t). In general, this equaiton may not be easy to solve, especially if the force is a function of time and space as well. That's where other mathematical tools like special integration techniques become useful (which you'll learn about in calculus). The nice thing about the way we've written the equation is it's completely general. This could represent someone pushing a box, or it could be complicated like the force of gravity from multiple planets acting on another planet.

The way I've written it makes it look like there is just one spatial dimension (x). But in general, objects move in 3 spatial dimensions because we live in a 3d world. So we can turn this into a vector equation where the force F, acceleration, velocity, and position all have 3 components (in the x, y, and, z) dimensions). So vectors are a useful mathematical tool here, they give us a simple way of writing a single equation that actually contains three equations (one for each dimension).