Numerical 1: Projectile from a Cliff

Problem: A stone is projected horizontally with $u = 15$ m/s from the top of a cliff of height $h = 45$ m. Find (a) time to reach the ground and (b) horizontal range. (g = 10 m/$s^{2}$)

Given:

• Initial horizontal velocity: $u_x = 15$ m/s
• Initial vertical velocity: $u_y = 0$ m/s (horizontal projection)
• Height: $h = 45$ m
• $g = 10$ m/$s^{2}$

Step 1 — Find time of flight (vertical motion only):

$s = u_y t + \tfrac{1}{2}g t^2$ $45\ \text{m} = (0\ \text{m/s})(t) + \tfrac{1}{2}(10\ \text{m/s}^2)(t^2)$ $45\ \text{m} = 5\ \text{m/s}^2 \times t^2$ $t^2 = \frac{45\ \text{m}}{5\ \text{m/s}^2} = 9\ \text{s}^2$ $\boxed{t = 3\ \text{s}}$

Step 2 — Find horizontal range:

$R = u_x \times t = 15\ \text{m/s} \times 3\ \text{s} = \boxed{45\ \text{m}}$

Dimensional check: $[LT^{-1}][T] = [L]$ ✓