Motion in 2 and 3 dimensions

Projectile Motion

Projectile motion is an example of motion along a curved path under the influence of a constant acceleration. It can be analysed as a two-dimensional problem.

A projectile is an object that is thrown obliquely into the air, and then follows a path determined entirely by the effects of gravity and air resistance. We shall assume here that the effects of air resistance can be neglected.

appletClick here for an interactive demonstration of projectile motion.

We can analyse projectile motion by treating the motion in the $x$ and $y$ directions separately.  The $x$–component of the acceleration is zero. The $y$–component is equal to $-g$ (choosing the positive $y$ direction to be upward). Therefore projectile motion can be considered as a combination of horizontal motion with constant velocity and vertical motion with constant acceleration.

We express all the vector relationships for position, velocity, and acceleration by separate equations for the horizontal and vertical components. The components of $\vec a$ are $$a_x=0,\quad a_y=-g\,.$$ Since the $x$–acceleration and the $y$–acceleration are both constant we can use the equations the equation for linear motion under constant acceleration for each separately. 

Considering the motion in the $x$ direction first, we have $$\begin{align*} v_x&=v_{x0},\\ x&=x_0+v_{x0}t\,. \end{align*}$$ For the motion in the $y$ direction where the acceleration is non-zero, $$\begin{align*} v_{y}&=v_{y0}-gt\,,\\ y&=y+0+v_{y0}t-\frac{1}{2}gt^2\,. \end{align*}$$For simplicity we now take the initial position of the projectile to be at the origin of our coordinate system, i.e., $x_0=y_0=0$.

projectile motion

The path of the projectile, and at a number of strategically chosen points we show the velocity, velocity components and acceleration.
Note: the $x$–component of acceleration is zero. Therefore $v_x$ is a constant; the $y$-component of acceleration is a constant (not zero) and therefore $v_y$ increases by equal amounts in equal increments of time.

Parabola

Can you show that the shape of the projectile's path (its trajectory) is of the form $y=bx-c x^2$, in other words, that it is of parabolic shape?

solutionClick here for a detailed solution.

Further questions

What is the skier's acceleration as she flies off the ramp (neglecting air resistance)?

skier on a ramp

Click on the image above to check your answer.