A ball is projected upwards from the top of a tower with a velocity `50ms^-1` making an angle `30^@` with the horizontal. The height of tower is 70m. After how many seconds from the instant of throwing, will the ball reach the ground. `(g=10 ms^-2)`

Options

A. 2 s
B. 5 s
C. 7 s
D. 9 s

Explanation

To solve the problem of a ball projected upwards from the top of a tower, we will break down the steps systematically.

### Step 1: Determine the initial velocity components
The ball is projected with an initial velocity \( u = 50 \, \text{m/s} \) at an angle of \( 30^\circ \) with the horizontal. We need to find the vertical component of the initial velocity \( u_y \).

\[ u_y = u \cdot \sin(30^\circ) = 50 \cdot \frac{1}{2} = 25 \, \text{m/s} \]

### Step 2: Set up the equation of motion
The height of the tower is \( h = 70 \, \text{m} \). When the ball reaches the ground, its vertical displacement \( s_y \) will be \( -70 \, \text{m} \) (since it falls downwards). The acceleration due to gravity \( g = 10 \, \text{m/s}^2 \) acts downwards, so it will be negative in our equation.

Using the equation of motion:
\[ s_y = u_y \cdot t + \frac{1}{2} a_y \cdot t^2 \]
Substituting the values:
\[ -70 = 25t - \frac{1}{2} \cdot 10 \cdot t^2 \]
This simplifies to:
\[ -70 = 25t - 5t^2 \]
Rearranging gives:
\[ 5t^2 - 25t - 70 = 0 \]

### Step 3: Solve the quadratic equation
To solve the quadratic equation \( 5t^2 - 25t - 70 = 0 \), we can use the quadratic formula:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 5 \), \( b = -25 \), and \( c = -70 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-25)^2 - 4 \cdot 5 \cdot (-70) = 625 + 1400 = 2025 \]

Now substituting into the quadratic formula:
\[ t = \frac{25 \pm \sqrt{2025}}{10} \]
Calculating \( \sqrt{2025} = 45 \):
\[ t = \frac{25 \pm 45}{10} \]
This gives us two possible solutions:
1. \( t = \frac{70}{10} = 7 \, \text{s} \)
2. \( t = \frac{-20}{10} = -2 \, \text{s} \) (not physically meaningful)

### Conclusion
The time after which the ball will reach the ground is \( t = 7 \, \text{s} \).