Applied Maths: Projectiles (Strand 3) — Higher Level Notes

1. The one idea the whole chapter rests on

Horizontal and vertical motion are independent. Gravity acts vertically, so it changes the vertical component and leaves the horizontal one completely alone:

a = 0i - gj

That is why a ball rolled off a table and a ball dropped beside it hit the floor at the same moment. Both start with zero vertical velocity and fall the same height. Horizontal speed has no say in it.

2. The four working equations

Resolve the projection velocity once, then treat each direction separately:

u = u cosα i + u sinα j

• Horizontal velocity is constant: v_x = u cosα
• Horizontal displacement: s_x = (u cosα)t
• Vertical velocity: v_y = u sinα - gt
• Vertical displacement: s_y = (u sinα)t - ½gt²

At the highest point the vertical component is zero — but the particle is still moving, horizontally, at u cosα. Saying the velocity is zero at the top is one of the most common errors on the paper.

3. Time of flight, range and greatest height

On level ground the standard results are:

T = 2u sinα / g   |   R = u² sin2α / g   |   H = u² sin²α / 2g

Note the 2α in the range. Using sinα instead halves your answer, and it is an easy thing to write without noticing.

4. Maximum range, and the two-angle result

Range depends on sin2α, which peaks when 2α = 90 degrees, so α = 45 degrees gives the greatest range for a given speed. Ninety degrees gives the greatest height and a range of zero.

Because sin2α = sin(180 - 2α), two complementary angles give the same range. Fire at 30 degrees or at 60 degrees and the ball lands in the same place — one on a flat path, one on a steep one. So when a question asks for "the two possible angles", finding one and stopping loses half the marks.

5. Questions that become a quadratic

When a projectile must pass through a named point, substitute the coordinates into the two displacement equations, eliminate t, and you are left with a quadratic — often in tanα. Both roots are usually genuine angles. Report both.

6. What is actually constant

Through the whole flight, the only thing that never changes is the acceleration — g downwards. The speed changes, the kinetic energy changes, the vertical velocity changes, the direction changes. The path is a parabola, because constant horizontal speed with uniform vertical acceleration makes y quadratic in x.

And mass cancels out of every equation above. With air resistance neglected, a heavy ball and a light one thrown identically land in exactly the same place.