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**Level Range Formulas**for projectile returning to its original height:

- horizontal range: $d_x = \frac{v_i^2 \cdot sin2\theta}{g}$
- time of flight: $t=\frac{2v_i \cdot sin\theta }{g}$

Examples

**Type 1 – Projectile at an Upward Angle**

A projectile is launched from the edge of a cliff 100 m above ground level with an initial speed of 20 m/s at an angle of 35° above the horizontal. Find:

- the horizontal and vertical components of the launch speed.
*the velocity vector at the maximum height.**the maximum height above the cliff top reached by the projectile.**the time taken to reach the maximum height.*- the projectile's time of flight.
- the projectile's range.
- the projectile's impact velocity (magnitude and direction) with the ground.

- the horizontal and vertical components of the launch speed.
**Type 2 – Projectile at a Downward Angle**

A hammer slides down a roof sloped at 35° reaching a speed of 6.2 m/s before falling off.

**Type 3 – Horizontally Launched Projectile**

A ball rolls off a cliff at 8 m/s and hits the ground with a speed of 25 m/s.

**Projectile Returning to the Same Height – Level Range Formulas**

Prove:*Level Range Formulas*- time of flight: $t=\frac{2v_i \cdot sin\theta }{g}$
- horizontal range: $d_x = \frac{v_i^2 \cdot sin2\theta}{g}$

A football is kicked at an angle 65° with a velocity of 18 m/s. Determine the football's range and time of flight: