An object is thrown vertically upwards and rises to a height of $10 \, m$. Calculate:
$(i)$ the velocity with which the object was thrown upward and
$(ii)$ the time taken by the object to reach the highest point.

(A) Given:
Height of the object,$h = 10 \, m$
Final velocity at the highest point,$v = 0 \, m/s$
Acceleration due to gravity,$a = -g = -9.8 \, m/s^2$
$(i)$ Using the third equation of motion,$v^2 - u^2 = 2ah$:
$0^2 - u^2 = 2 \times (-9.8) \times 10$
$-u^2 = -196$
$u^2 = 196$
$u = 14 \, m/s$
Thus,the initial velocity is $14 \, m/s$.
$(ii)$ Using the first equation of motion,$v = u + at$:
$0 = 14 + (-9.8) \times t$
$9.8t = 14$
$t = 14 / 9.8 \approx 1.43 \, s$
Thus,the time taken to reach the highest point is approximately $1.43 \, s$.