A body is projected with velocity 24 m/s making an angle 30 degrees with the horizontal. The vertical component of its velocity after 2 s is (g = 10 m/s^2): A) 8 m/s upward B) 8 m/s downward C) 32 m/s upward D) 32 m/s downward
Question
A body is projected with velocity $24 \text{ m/s}$ making an angle $30^\circ$ with the horizontal. The vertical component of its velocity after $2 \text{ s}$ is ($\text{g} = 10 \text{ m/s}^2$):
A) $8 \text{ m/s}$ upward
B) $8 \text{ m/s}$ downward
C) $32 \text{ m/s}$ upward
D) $32 \text{ m/s}$ downward
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Answer
To find the vertical component of the velocity of a body projected with an initial velocity of $24 \text{ m/s}$ at an angle of $30^\circ$ with the horizontal after $2 \text{ s}$, given that the acceleration due to gravity $\text{g} = 10 \text{ m/s}^2$, follow these steps:

Resolve the initial velocity into its vertical component:
 The initial velocity (v = 24 \text{ m/s})
 The angle of projection (\theta = 30^\circ)
 The vertical component of the initial velocity is given by: $$ v_{y0} = v \sin \theta = 24 \sin 30^\circ $$
 Since (\sin 30^\circ = 0.5): $$ v_{y0} = 24 \times 0.5 = 12 \text{ m/s} $$

Use the kinematic equation to find the vertical component of velocity after (2 \text{ s}):
 The kinematic equation is: $$ v_y = v_{y0} + a \cdot t $$
 Here, (v_y) is the final vertical velocity, (v_{y0} = 12 \text{ m/s}) is the initial vertical velocity, (a = g = 10 \text{ m/s}^2) (since gravity acts downwards), and (t = 2 \text{ s}): $$ v_y = 12 \text{ m/s} + (10 \text{ m/s}^2) \times 2 \text{ s} $$ $$ v_y = 12 \text{ m/s}  20 \text{ m/s} $$ $$ v_y = 8 \text{ m/s} $$

Interpret the result:
 The negative sign indicates that the vertical component of the velocity is downward.
Thus, after 2 seconds, the vertical component of the velocity is 8 m/s downward.
Final Answer: B) (8 \text{ m/s}) downward
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