A ball is thrown vertically upwards from the ground and a student gazing out of the window sees it moving upward past him at `10ms^-1.` The window is at 15 m above the ground level. The velocity of ball 3 s after it was projected from the ground is [Take `g= 10 ms^-2`]

Options

A. `10 m//s,` up
B. `20 m//s,` up
C. `20 ms^-1,`down
D. `10 ms^-1,` down

Explanation

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the problem
A ball is thrown vertically upwards from the ground, and a student sees it moving upward past a window at a speed of 10 m/s. The window is 15 m above the ground. We need to find the velocity of the ball 3 seconds after it was projected.

### Step 2: Identify the known values
- Velocity of the ball as seen by the student (v) = 10 m/s (upward)
- Height of the window (s) = 15 m
- Acceleration due to gravity (g) = 10 m/s² (downward)
- We need to find the initial velocity (u) of the ball when it was thrown.

### Step 3: Use the kinematic equation
We can use the kinematic equation:
\[ v^2 = u^2 + 2as \]
Where:
- v = final velocity (10 m/s)
- u = initial velocity (unknown)
- a = acceleration (-g = -10 m/s², since it acts downward)
- s = displacement (15 m, upward)

### Step 4: Substitute the known values into the equation
Substituting the known values into the equation:
\[ (10)^2 = u^2 + 2(-10)(15) \]
\[ 100 = u^2 - 300 \]

### Step 5: Solve for u
Rearranging the equation gives:
\[ u^2 = 100 + 300 \]
\[ u^2 = 400 \]
Taking the square root:
\[ u = \sqrt{400} = 20 \text{ m/s} \]

### Step 6: Find the velocity after 3 seconds
Now we will find the velocity of the ball after 3 seconds using the equation:
\[ v = u + at \]
Where:
- u = 20 m/s (initial velocity)
- a = -10 m/s² (acceleration due to gravity)
- t = 3 s

### Step 7: Substitute the values into the equation
Substituting the values:
\[ v = 20 + (-10)(3) \]
\[ v = 20 - 30 \]
\[ v = -10 \text{ m/s} \]

### Step 8: Interpret the result
The negative sign indicates that the velocity is directed downward. Therefore, the velocity of the ball after 3 seconds is 10 m/s downward.

### Final Answer
The velocity of the ball 3 seconds after it was projected is **10 m/s downward**.

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