Answer:
The maximum height it reaches is 20 meters
The time taken by it to reach the height is 2 seconds
Step-by-step explanation:
The formula of the height of the stone is h = u t - [tex]\frac{1}{2}[/tex] g t², where
- t is the time to reach the height h
- u is the initial velocity
- g is the acceleration of gravity
∵ A stone is thrown vertically upwards with an initial velocity of 20 m/s
∴ u = 20 m/s
∵ g = 10 m/s²
→ Substitute them in the equation above
∴ h(t) = 20t - [tex]\frac{1}{2}[/tex] (10) t²
∴ h(t) = 20t - 5t²
→ Arrange the terms of the right side according to the greatest power of t
∴ h(t) = -5t² + 20t
To find the maximum height and the time of it find the vertex of the quadratic function (m, k), where m = [tex]\frac{-b}{2a}[/tex] and k is the value of h at t = m, a is the coefficient of t² and b is the coefficient of t
∵ The coefficient of t² is -5
∴ a = -5
∵ The coefficient of t is 20
∴ b = 20
→ Use them to find h
∵ m = [tex]\frac{-20}{2(-5)}[/tex] = [tex]\frac{-20}{-10}[/tex] = h
∴ m = 2
→ Substitute it in the equation above to find k
∵ h(m) = k
∵ k = -5(2)² + 20(2)
∴ k = -5(4) + 40
∴ k = -20 + 40
∴ k = 20
∴ The coordinate of the vertex of the function are (2, 20)
→ m represents the time of the maximum height and k represents
the maximum height
∴ The maximum height it reaches is 20 meters
∴ The time taken by it to reach the height is 2 seconds
