Projection at an angle
G1.4
By the end of this lesson you should be able to
- Solve problems involving projectile motion in the absence of drag effects using v_y = u_y + gt, s_y = u_y t + ½gt^2, v_y^2 = u_y^2 + 2gs_y, v_x = u_x and s_x = u_x t. G1.4
Key ideas
- Start by resolving the launch velocity: \(u_x = u\cos\theta\), \(u_y = u\sin\theta\).
- At maximum height the vertical velocity is zero; the horizontal velocity is unchanged.
- On level ground the flight is symmetrical: time up = time down, and landing speed = launch speed.
- Use \(v_y = 0\) at the top to find time to the top and maximum height; double the time for the total flight (level ground only).
Key formulas (QCAA formula book notation)
\[
v_y = u_y + g t
\]
\[
s_y = u_y t + \tfrac{1}{2} g t^2
\]
\[
v_y^2 = u_y^2 + 2 g s_y
\]
\[
s_x = u_x t
\]
Worked example
A ball is kicked at 20 m s⁻¹ at 35° from level ground. Find time of flight, range and maximum height.
- \(u_x = 16.4\) m s⁻¹, \(u_y = 11.5\) m s⁻¹
- Time to top: \(t = 11.5/9.8 = 1.17\) s, so flight = 2.34 s
- Range: \(16.4 \times 2.34 = 38.4\) m
- Max height: \(h = 11.5^2/(2 \times 9.8) = 6.74\) m
Common mistakes
- Using the full launch speed instead of a component.
- Doubling the time when launch and landing heights differ.
- Saying the velocity is zero at the top (only the vertical component is).
Do
- Read: Textbook (Oxford Physics for Queensland Units 3 & 4) section 2.3
- Do: CYL 2.3 Q1, 4, 5, 8
- Self-check: Projectile motion self-check
Try it
PhET: Projectile Motion. Predict the time of flight and range first, then fire and use the measuring tool to check.
Open it in a new tab if it does not load.
Watch
- Nerd-A-Pult: An Introductory Projectile Motion Problem · Flipping Physics
Further reading
- OpenStax Physics 5.3 Projectile motion (OpenStax, CC BY 4.0)
- Full topic notes: Gravity & Motion