Physics / Unit 2: Vectors and Motion in Two Dimensions
2.4 Projectiles Launched at an Angle
Notes
For a launch at speed $v$ and angle $\theta$, split the velocity: $v_x = v\cos\theta$ and $v_y = v\sin\theta$.
- $v_x$ stays constant the whole flight; $v_y$ decreases by $9.8 \text{ m/s}$ each second.
- At the top, $v_y = 0$ but $v_x$ is not zero.
- On level ground: time up $\displaystyle = \frac{v_y}{g}$, total time $\displaystyle = \frac{2v_y}{g}$, range $= v_x \times$ total time.
- Without air, a $45^\circ$ launch gives the greatest range on level ground; $30^\circ$ and $60^\circ$ give equal ranges.
- Example: $v_y = 14.7 \text{ m/s}$ gives time up $= 1.5 \text{ s}$ and total time $3 \text{ s}$.
Practice
At the highest point of a soccer ball's flight, what is its velocity?
Vertical velocity is zero at the top, but the horizontal part stays the same.
A ball is launched with $v_x = 12 \text{ m/s}$ and $v_y = 19.6 \text{ m/s}$ on level ground. What is its range?
Total time $\displaystyle = \frac{2(19.6)}{9.8} = 4 \text{ s}$; range $= 12 \times 4 = 48 \text{ m}$.
Ignoring air, which launch angle gives the same range as $25^\circ$ at the same speed?
Complementary angles (adding to $90^\circ$) give equal ranges on level ground.
Ready for more? Take the Unit 2 test.