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:

  1. 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): $$ v_{y0} = 24 \times 0.5 = 12 \text{ m/s} $$
  2. 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}): $$ v_y = 12 \text{ m/s} + (-10 \text{ m/s}^2) \times 2 \text{ s} $$ $$ v_y = 12 \text{ m/s} - 20 \text{ m/s} $$ $$ v_y = -8 \text{ m/s} $$
  3. 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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