Projectile motion describes any object moving through the air under the influence of gravity alone, following a curved path called a parabola. From a thrown ball to a launched rocket, the same equations govern the flight. The key insight is that horizontal and vertical motion are independent: gravity pulls the object downward at a constant rate while the horizontal velocity stays constant (ignoring air resistance).
The launch angle determines the shape of the trajectory. At 45 degrees you get maximum range on flat ground. Lower angles produce flatter, shorter flights; higher angles send the projectile higher but bring it back sooner.
Understanding these relationships lets you predict exactly where and when a projectile will land.
Understanding Projectile Motion
A projectile problem becomes easier when you treat the launch as two separate starting motions. The initial speed is split into a sideways part and an upward part. A shallow launch gives most of its speed sideways.
A steep launch gives more speed upward. These parts are found using sine and cosine, so a calculator must be in degree mode when the angle is given in degrees. Draw a small right triangle beside the launch velocity.
The side next to the angle gives the horizontal part, while the opposite side gives the vertical part. This sketch prevents many sign and component mistakes.
At the highest point, the vertical velocity is zero for one instant. The object is not motionless there because it still has sideways velocity. Gravity continues acting at that point, so the vertical velocity immediately becomes downward.
This is why an object rises more slowly, pauses briefly, then gains downward speed. On a vertical velocity against time graph, this motion makes a straight line with a negative slope. The slope is set by gravitational field strength.
Near Earth’s surface, this is about 9.8 metres per second squared downward. In many school calculations it is rounded to 9.8 or 10 metres per second squared, so use the value stated in the question.
The familiar range and flight time results only work directly when the projectile lands at the same height from which it was launched. A football kicked from a balcony, a stone thrown into a pit, or a ball rolling off a table needs a different approach. First find the time from the vertical motion using the starting height and the final height.
Then use that same time in the horizontal motion to find how far it travels. This shared time links the two directions.
Choose upward as positive or downward as positive, then keep that choice throughout the calculation. Mixing signs is a common reason for impossible negative times or heights.
Real projectiles are affected by air resistance, wind, spin, and changes in gravity, though basic questions usually leave these out. Air resistance acts opposite to the direction of travel. It slows the horizontal motion and makes the downward path steeper than the upward path.
Spin can bend a ball through the Magnus effect, which helps explain curling football shots and some tennis serves. A realistic trajectory is therefore not perfectly symmetrical. When studying textbook problems, read the assumptions carefully.
Words such as level ground, neglect air resistance, and launched from rest at a height tell you which model applies. Include units at every step.
Speeds use metres per second, time uses seconds, distances use metres, and acceleration uses metres per second squared. Unit checks often reveal an error before the final answer.
Key Facts
- Horizontal velocity is constant:
- Vertical velocity changes due to gravity:
- Time of flight:
- Maximum height:
- Horizontal range:
- Range is the same for complementary angles (30° and 60° give equal range)
Vocabulary
- Trajectory
- The curved path followed by a projectile through the air.
- Launch angle
- The angle between the initial velocity vector and the horizontal.
- Range
- Horizontal distance traveled from launch to landing on the same level.
- Apex
- The highest point in the projectile's path where vertical velocity equals zero.
- Time of flight
- Total time the projectile is in the air from launch to landing.
Common Mistakes to Avoid
- Treating horizontal and vertical motion as linked - they are independent. Horizontal speed does not change; only vertical speed changes due to gravity.
- Using g = 9.8 m/s² as positive when setting up vertical equations. Choose a sign convention and stick to it: if up is positive, g should be negative in your equations.
- Forgetting that maximum range on flat ground occurs at exactly 45 degrees only when the launch and landing heights are equal.
- Ignoring the two solutions for time when solving quadratic equations - the negative root represents a time before launch and should be discarded.
Practice Questions
- 1 A ball is kicked at 20 m/s at an angle of 30°. How long is it in the air and how far does it travel horizontally?
- 2 A projectile reaches a maximum height of 45 m. What was its initial vertical velocity?
- 3 Two projectiles are launched at 15° and 75° from the same point with the same initial speed. Without calculating, explain why they travel the same horizontal range.