A projectile is thrown upward with a velocity `v_(0)` at an angle `alpha` to the horizontal. The change in velocity of the projectile when it strikes the same horizontal plane is

Options

A. `v_(0) sin alpha` vertically downward
B. `2v_(0) sin alpha` vertically downward
C. `2v_(0) sin alpha` vertically upward
D. zero

Explanation

To solve the problem of finding the change in velocity of a projectile when it strikes the same horizontal plane, we can follow these steps:

### Step 1: Understand the Initial Velocity Components
When the projectile is thrown with an initial velocity \( v_0 \) at an angle \( \alpha \) to the horizontal, we can break this velocity into its horizontal and vertical components:
- The horizontal component \( v_{0x} = v_0 \cos \alpha \)
- The vertical component \( v_{0y} = v_0 \sin \alpha \)

### Step 2: Determine the Final Velocity Components
When the projectile returns to the same horizontal level, its horizontal component of velocity remains unchanged (assuming no air resistance):
- The horizontal component \( v_{fx} = v_0 \cos \alpha \)

The vertical component of the velocity when the projectile strikes the ground will be equal in magnitude but opposite in direction to the initial vertical component:
- The final vertical component \( v_{fy} = -v_0 \sin \alpha \) (downward)

### Step 3: Calculate the Change in Velocity
The change in velocity \( \Delta v \) can be calculated by subtracting the initial velocity vector from the final velocity vector:
\[ \Delta v = \vec{v_f} - \vec{v_i} \]
Where:
- \( \vec{v_i} = v_{0x} \hat{i} + v_{0y} \hat{j} = v_0 \cos \alpha \hat{i} + v_0 \sin \alpha \hat{j} \)
- \( \vec{v_f} = v_{fx} \hat{i} + v_{fy} \hat{j} = v_0 \cos \alpha \hat{i} - v_0 \sin \alpha \hat{j} \)

Now, substituting these into the change in velocity equation:
\[ \Delta v = (v_0 \cos \alpha \hat{i} - v_0 \sin \alpha \hat{j}) - (v_0 \cos \alpha \hat{i} + v_0 \sin \alpha \hat{j}) \]

### Step 4: Simplify the Change in Velocity
Simplifying the equation:
\[ \Delta v = v_0 \cos \alpha \hat{i} - v_0 \sin \alpha \hat{j} - v_0 \cos \alpha \hat{i} - v_0 \sin \alpha \hat{j} \]
\[ \Delta v = -2v_0 \sin \alpha \hat{j} \]

### Step 5: Interpret the Result
The change in velocity is \( -2v_0 \sin \alpha \hat{j} \), indicating that the change in velocity is directed downward.

### Final Answer
Thus, the change in velocity of the projectile when it strikes the same horizontal plane is:
\[ \Delta v = 2v_0 \sin \alpha \text{ vertically downward} \]