A body is projected with velocity 24 `ms^(-1)` making an angle 30° with the horizontal. The vertical component of its velocity after 2s is (g=10`ms^(-1)` )

Options

A. 8 `ms^(-1)` upward
B. 8 `ms^(-1)` down ward
C. 32 `ms^(-1)` upward
D. 32 `ms^(-1)` down ward

Answer

To solve the problem step by step, we will follow the physics of projectile motion. The goal is to find the vertical component of the velocity of a body projected at a certain angle after a specific time.

### Step 1: Identify the initial velocity and angle
The body is projected with an initial velocity \( v = 24 \, \text{ms}^{-1} \) at an angle \( \theta = 30^\circ \) with the horizontal.

### Step 2: Calculate the vertical component of the initial velocity
The vertical component of the initial velocity \( v_{y0} \) can be calculated using the formula:
\[ v_{y0} = v \sin \theta \]
Substituting the values:
\[ v_{y0} = 24 \sin(30^\circ) \]
Since \( \sin(30^\circ) = \frac{1}{2} \):
\[ v_{y0} = 24 \times \frac{1}{2} = 12 \, \text{ms}^{-1} \]

### Step 3: Apply the kinematic equation for vertical motion
The vertical velocity \( v_y \) after time \( t \) can be calculated using the equation:
\[ v_y = v_{y0} - g t \]
where \( g = 10 \, \text{ms}^{-2} \) (acceleration due to gravity) and \( t = 2 \, \text{s} \).

### Step 4: Substitute the values into the equation
Substituting \( v_{y0} = 12 \, \text{ms}^{-1} \), \( g = 10 \, \text{ms}^{-2} \), and \( t = 2 \, \text{s} \):
\[ v_y = 12 - 10 \times 2 \]
\[ v_y = 12 - 20 \]
\[ v_y = -8 \, \text{ms}^{-1} \]

### Step 5: Interpret the result
The negative sign indicates that the vertical component of the velocity is directed downwards. Thus, the vertical component of the velocity after 2 seconds is:
\[ 8 \, \text{ms}^{-1} \text{ downward} \]

### Final Answer
The vertical component of its velocity after 2 seconds is \( 8 \, \text{ms}^{-1} \) downward.

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