The equation of trajectory of a projectile is given as y = 2x - (x^2)/2. The maximum height of the projectile is (Symbols have usual meanings and SI unit):
(1) 4 m
(2) 1 m
(3) 2 m
(4) 10 m
Answer (1)
To find the maximum height of the projectile, we need to analyze the trajectory equation given as y = 2 x − 2 x 2 .
This equation is in the form of a quadratic equation y = a x 2 + b x + c where a = − 2 1 , b = 2 , and c = 0 .
The trajectory equation can be rearranged into a standard quadratic form: y = − 2 1 x 2 + 2 x
In a quadratic equation of the form y = a x 2 + b x + c , the vertex represents the maximum or minimum point of the parabola. Since the coefficient a is negative ( − 2 1 ), the parabola opens downwards, indicating that the vertex will give the maximum point of the parabola, or the maximum height of the projectile.
The x-coordinate of the vertex can be found using the formula: x = − 2 a b
Substitute the values of a and b :
x = − 2 × − 2 1 2 = − − 1 2 = 2
Now, substitute x = 2 back into the trajectory equation to find the maximum height y :
y = 2 ( 2 ) − 2 ( 2 ) 2 y = 4 − 2 4 y = 4 − 2 = 2
Therefore, the maximum height of the projectile is 2 meters.
Based on the choices provided, the correct option is:
(3) 2 m
