NSW HSC Physics

Two independent motions

A projectile is an object moving under the influence of gravity alone after it has been launched. Its path is curved, but the curve is easy to analyse because the motion splits into a horizontal part and a vertical part that do not affect each other. The force of gravity acts straight down, so it changes only the vertical velocity. Nothing pushes sideways, so the horizontal velocity stays as it was.

The analysis rests on two assumptions.

• The vertical acceleration is constant and equal to the acceleration due to gravity, $g = 9.8\,\mathrm{m\,s^{-2}}$ downward. This is reasonable when the height gained is small compared with the radius of the Earth.
• There is no air resistance. In reality drag shortens the range and lowers the maximum height, especially for light objects and high speeds, so the predictions are upper limits for a real projectile.

Choosing the launch point as the origin, with up as positive, a projectile launched with speed $u$ at an angle $\theta$ above the horizontal has initial velocity components

$u_x$ and $u_y$ are the horizontal and vertical components in metres per second, and $\theta$ is measured from the horizontal.

Equations for each direction

The horizontal motion is at constant velocity, and the vertical motion is at constant acceleration $a_y = -g$.

These are the equations of uniformly accelerated motion, $v = u + at$, $s = ut + \tfrac12 at^2$, $v^2 = u^2 + 2as$ and $s = \tfrac12(u+v)t$, with the vertical quantities put in. Time is the one quantity that links the two directions: the same $t$ is used in the horizontal and the vertical equations, so a typical problem finds the time from the vertical motion and then uses it to find how far the projectile travels sideways.

Velocity at any instant

At any moment the velocity is the vector sum of $v_x$ and $v_y$, so its magnitude and direction are

where $\phi$ is the angle of the velocity below or above the horizontal. The speed is smallest at the top of the flight, where $v_y = 0$ and only $u_x$ remains. On the way up the vertical velocity is positive and shrinking; on the way down it is negative and growing in magnitude, while the horizontal velocity never changes.

Features of the path

Because the vertical acceleration is constant and the horizontal velocity is constant, eliminating $t$ gives $y$ as a quadratic in $x$, so the path is a parabola. For a projectile launched from the origin,

Other features follow from the symmetry of the parabola when the projectile lands at the height it was launched from.

• The time to reach maximum height is half the total time of flight.
• The speed on landing equals the launch speed, and the landing angle below the horizontal equals the launch angle above it.
• At maximum height the velocity is horizontal and the acceleration is still $g$ downward. The acceleration is never zero, not even at the top.

The symmetry fails when the landing height differs from the launch height, which is dealt with for launches from a height in the work on range and maximum height.

Worked example

A ball is kicked from level ground at $25\,\mathrm{m\,s^{-1}}$ at $40^\circ$ above the horizontal. Find the time to reach the highest point, the maximum height, the time of flight and the horizontal distance travelled before it lands.

The components of the launch velocity are

At the highest point $v_y = 0$, so $0 = u_y - gt$ and

The height gained comes from $v_y^2 = u_y^2 - 2gy$ with $v_y = 0$:

The ball lands at the launch height, so the time of flight is twice the time up, $3.28\,\mathrm{s}$. The horizontal distance in that time is

Comparing projectiles

Two objects released from the same height, one dropped and one thrown horizontally, reach the ground together, because the sideways velocity has no effect on the vertical motion. A horizontal launch is simply the case $\theta = 0$, where $u_y = 0$ and the vertical motion is a fall from rest.

The same approach works for any situation where the net force on the object is constant, which is why the motion of a charged particle in a uniform electric field is treated in the same way, with the electric force taking the place of weight.