Question

The minimum speed in m/s with which a projectile must be thrown from origin at ground so that it is able to pass through a point P (30 m, 40 m) is ______. (g = 10 m/s²)

Options

• 10

• 20

• 30

• 40

Answer 1

The minimum speed in m/s with which a projectile must be thrown from origin at ground so that it is able to pass through a point P (30 m, 40 m) is 30. (g = 10 m/s²)

Explanation:

Let the velocity of the projectile be u co-ordinate of the point P = (30 m, 40 m)

Trajector of the projectile

y = x tan θ - `1/2 "g"  x^2/("u"^2costheta)`

Now, y = 40, x = 30

Substituting the values 

40 = 30 tan θ - `1/2xx10xx30^2/(4^2cos^2 theta)`

40 = 30 tan θ - `(10xx900)/(2"u"^2)(1+tan^2theta)`

 ⇒ 8u² = 2 × 3u² tan θ – 900 (1 + tan² θ)

⇒ 900 tan² θ – 6u² tan θ + (8u² + 900) = 0

For real value of θ,

if b² – 4ac ≥ 0

⇒ (6u²)² ≥ 4 × 900 × (8u² + 900)

⇒  36u⁴ ≥ 3600 (8u² + 900)

⇒ u⁴ ≥ 100(8u² + 900)

⇒ u⁴ ≥ 800u² + 90000

⇒ u⁴ – 800u² ≥ 90000

⇒ u⁴ – 2 × 400u² + 400² ≥ 250000

⇒ (u² – 400)² ≥ 250000

⇒ u² – 400 ≥ 500

⇒ u² ≥ 900

⇒ u ≥ 30 m/s