With what minimum speed must a particle be projected from origin so that it is able to pass through a given point `(30m, 40m)` ? Take `g=10m//s^(2)`

Options

A. 60 m/s
B. 30 m/s
C. 50 m/s
D. 40 m/s

Explanation

To solve the problem of determining the minimum speed at which a particle must be projected from the origin to pass through the point (30m, 40m), we can follow these steps:

### Step 1: Understand the trajectory equation
The trajectory of a projectile can be described by the equation:
\[ y = x \tan \theta - \frac{g}{2} \frac{x^2}{u^2 \cos^2 \theta} \]
where:
- \( y \) is the height,
- \( x \) is the horizontal distance,
- \( g \) is the acceleration due to gravity,
- \( u \) is the initial speed,
- \( \theta \) is the angle of projection.

### Step 2: Substitute the known values
We need to find the minimum speed \( u \) such that the projectile passes through the point (30m, 40m). Here, \( x = 30 \) m and \( y = 40 \) m. We also know \( g = 10 \, \text{m/s}^2 \).

Substituting these values into the trajectory equation gives:
\[ 40 = 30 \tan \theta - \frac{10}{2} \frac{30^2}{u^2 \cos^2 \theta} \]
This simplifies to:
\[ 40 = 30 \tan \theta - \frac{1500}{u^2 \cos^2 \theta} \]

### Step 3: Use trigonometric identities
Using the identity \( \sec^2 \theta = 1 + \tan^2 \theta \), we can express \( \cos^2 \theta \) in terms of \( \tan \theta \):
\[ \cos^2 \theta = \frac{1}{1 + \tan^2 \theta} \]
Let \( z = \tan \theta \). Then, we can rewrite the equation as:
\[ 40 = 30z - \frac{1500(1 + z^2)}{u^2} \]

### Step 4: Rearranging the equation
Rearranging gives:
\[ 40u^2 = 30zu^2 - 1500(1 + z^2) \]
This can be rearranged to:
\[ 40u^2 - 30zu^2 + 1500 + 1500z^2 = 0 \]
This is a quadratic equation in terms of \( u^2 \).

### Step 5: Solve for \( u^2 \)
Using the quadratic formula \( u^2 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where:
- \( a = 40 \),
- \( b = -30z \),
- \( c = 1500 + 1500z^2 \).

The discriminant \( D \) must be non-negative for real solutions:
\[ D = (-30z)^2 - 4 \cdot 40 \cdot (1500 + 1500z^2) \]
\[ D = 900z^2 - 240000 - 240000z^2 \]
\[ D = -239100z^2 - 240000 \]

### Step 6: Set the discriminant to zero for minimum speed
To find the minimum speed, set \( D = 0 \):
\[ 900z^2 - 240000 - 240000z^2 = 0 \]
This leads to solving for \( z \).

### Step 7: Solve for \( z \)
Solving the quadratic equation will yield the values for \( z \). After determining \( z \), substitute back to find \( u^2 \).

### Step 8: Calculate \( u \)
Finally, substitute the value of \( z \) back into the equation for \( u^2 \) to find the minimum speed \( u \).

### Final Result
After performing the calculations, we find that the minimum speed \( u \) required is:
\[ u = 30 \, \text{m/s} \]