Question

Can explain projectile moton

Answer 1

Sure — I think you mean "projectile motion." Here's a clear, compact explanation with the main formulas and a worked example.

What projectile motion is
- Motion of an object thrown or launched near the earth’s surface, moving under the influence of gravity only (air resistance neglected).
- Motion splits into two independent components: horizontal (constant velocity) and vertical (constant acceleration = gravity).

Set-up and notation
- Initial speed: u (or v0)
- Launch angle above horizontal: θ
- Horizontal component: ux = u cosθ
- Vertical component: uy = u sinθ
- Acceleration due to gravity: g ≈ 9.8 m/s² downward (use +g in formulas where appropriate)

Kinematic equations (taking origin at launch, upward positive)
- Horizontal position: x(t) = (u cosθ) t
- Vertical position: y(t) = (u sinθ) t − ½ g t²
- Horizontal velocity: vx = u cosθ (constant)
- Vertical velocity: vy(t) = u sinθ − g t

Key results (launch and landing at same height)
- Time of flight: T = 2u sinθ / g
- Maximum height: H = (u² sin²θ) / (2g)
- Range (horizontal distance): R = (u² sin 2θ) / g
- Trajectory equation (y vs x): y(x) = x tanθ − (g x²) / (2 u² cos²θ) (a parabola)

Important points
- Horizontal and vertical motions are independent.
- For given speed u on level ground, range is maximized when θ = 45° (because sin2θ is max).
- If launch and landing heights differ, solve the vertical quadratic for t (y(t) = final height) and use x = ux t to get range.
Example formula if launching from height h above landing level:
t = [u sinθ + sqrt((u sinθ)² + 2 g h)] / g (choose positive root)
then R = (u cosθ) t.

Worked example
- Let u = 20 m/s, θ = 30°.
ux = 20 cos30° = 17.32 m/s
uy = 20 sin30° = 10.0 m/s
Time of flight: T = 2·10 / 9.8 ≈ 2.04 s
Max height: H = (20² sin²30°) / (2·9.8) = (400·0.25) / 19.6 ≈ 5.10 m
Range: R = (20² sin60°) / 9.8 = 400·0.866 / 9.8 ≈ 35.3 m

Air resistance
- Neglecting it gives parabolic trajectories. With drag, motion is more complex (non-parabolic, shorter range) and usually requires numeric methods or approximations.

Want more?
- I can derive any formula step-by-step, solve a specific problem, or show graphs/plots of trajectories. Which would you like?