A pebble is thrown vertically upwards from a bridge with an initial velocity of 4.9 m/s. It strikes the water after 2s. If acceleration due to gravity is `9.8 m//s^(2)(a)` what is the height of the bridge? (b) with what velocity does that pebble strike the water?

Options

A. `(6H)/(T_(P)^(2)+T_(Q)^(2))`
B. `(8H)/(T_(P)^(2)+T_(Q)^(2))`
C. `(2H)/(T_(P)^(2)+T_(Q)^(2))`
D. `(H)/(T_(P)^(2)+T_(Q)^(2))`

Explanation

To solve the problem step-by-step, we will use the equations of motion.

### Given:
- Initial velocity of the pebble, \( u = 4.9 \, \text{m/s} \) (upward)
- Time of flight, \( t = 2 \, \text{s} \)
- Acceleration due to gravity, \( g = 9.8 \, \text{m/s}^2 \) (downward)

### (a) Finding the height of the bridge

1. **Use the second equation of motion**:
The second equation of motion is given by:
\[ s = ut + \frac{1}{2} a t^2 \]
Here, \( s \) is the displacement (height of the bridge), \( u \) is the initial velocity, \( a \) is the acceleration (which will be negative since we are considering upward direction as negative), and \( t \) is the time.

2. **Substituting values**:
Since we are taking downward as positive, we will consider \( a = -g = -9.8 \, \text{m/s}^2 \). The equation becomes:
\[ s = 4.9 \cdot 2 + \frac{1}{2} \cdot (-9.8) \cdot (2^2) \]

3. **Calculating the terms**:
- First term: \( 4.9 \cdot 2 = 9.8 \, \text{m} \)
- Second term: \( \frac{1}{2} \cdot (-9.8) \cdot 4 = -19.6 \, \text{m} \)

4. **Finding the total displacement**:
\[ s = 9.8 - 19.6 = -9.8 \, \text{m} \]
Since the height is measured from the bridge to the water, we take the absolute value:
\[ \text{Height of the bridge} = 9.8 \, \text{m} \]

### (b) Finding the velocity with which the pebble strikes the water

1. **Use the first equation of motion**:
The first equation of motion is given by:
\[ v = u + at \]
Here, \( v \) is the final velocity, \( u \) is the initial velocity, \( a \) is the acceleration, and \( t \) is the time.

2. **Substituting values**:
Again, taking downward as positive, we have:
\[ v = 4.9 + (-9.8) \cdot 2 \]

3. **Calculating the terms**:
\[ v = 4.9 - 19.6 = -14.7 \, \text{m/s} \]
The negative sign indicates that the direction of the velocity is downward. Thus, the magnitude of the velocity is:
\[ \text{Velocity with which it strikes the water} = 14.7 \, \text{m/s} \]

### Summary of Answers:
- (a) Height of the bridge = \( 9.8 \, \text{m} \)
- (b) Velocity with which the pebble strikes the water = \( 14.7 \, \text{m/s} \)