## 2D Kinematics

In AS Maths we saw how to use the SUVAT equations in one direction when acceleration is constant. We also learned about the calculus of position, velocity and acceleration in one direction when acceleration is not constant. In A2 Maths, we extend these ideas to two dimensions and apply the SUVAT equations to bodies moving under constant acceleration such as projectiles and perform calculus for those moving under variable acceleration.

### Constant Acceleration

We model a projectile as a particle acting under gravity only (we ignore air resistance etc) that is given some initial velocity at a given angle to the ground. Acceleration is zero in the horizontal direction and has gravitational acceleration in the vertical direction. Since acceleration is constant in both directions, we can resolve the initial velocity vector (as we did when resolving forces) and apply the SUVAT equations in the individual directions.

Questions could ask you to find the horizontal range of the particle (the horizontal distance it reaches before hitting the ground), the greatest height it reaches (found when vertical velocity is zero) or the time of flight – see Example 1.

When all of the information is given algebraically, it can be shown that, for a particle projected from the ground:

- Time of flight:
- Range:
- Equation of trajectory:

Note that is the value of acceleration here and not the vector . We see that the equation of trajectory is quadratic in and so, for a particle projected from the ground, we also have that the time to greatest height is half of time of flight. Questions may ask you to derive the formulae above – see Example 2.

For particles moving in a plane with constant velocity (), the position vector of the particle is given by where is the initial position of the particle. For particles moving with constant acceleration (), the particle has velocity and position noting the bold vectors and the non-bold scalars. You may recognise these as SUVAT equation for more than one dimension. Questions will likely require the use of the unit vectors and that we learned about in AS Maths – see Example 3.

### Variable Acceleration

Recall from AS Maths that when acceleration is variable then velocity can be found by either differentiating position or integrating acceleration – we took a look at the motion of a particle with variable acceleration moving along a straight line. Similarly, acceleration can be found by differentiating velocity or position can be found by integrating velocity. In more than one dimension, we have

and

noting the use of vectors to show that this is now in multiple dimensions. Note that sometimes the use of a dot is shorthand for differentiation with respect to time and that questions will again require the use of the unit vectors – see Example 4.