Question

A body is projected with velocity 24 m/s making an angle of 30 degrees with the horizontal. The vertical component of its velocity after 2 seconds is (g = 10 m/s^2):

A. 8 m/s upward
B. 8 m/s downward
C. Not given
D. 32 m/s downward

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Answer

To determine the vertical component of the velocity of a body projected with an initial velocity of 24 m/s at an angle of 30 degrees after 2 seconds, given $ g = 10 , \text{m/s}^2 $, we proceed as follows:

  1. Resolve the initial velocity into vertical and horizontal components: The vertical component of the initial velocity is given by: $$ v_y = v \sin \theta $$ Here, $ v = 24 , \text{m/s} $ and $ \theta = 30^\circ $. Using the fact that $ \sin 30^\circ = \frac{1}{2} $, we get: $$ v_y = 24 \times \frac{1}{2} = 12 , \text{m/s} $$

  2. Calculate the final vertical velocity using kinematic equations: The final vertical velocity can be calculated using the equation: $$ v_{fy} = v_{iy} + a t $$ Here, $ v_{iy} $ is the initial vertical velocity, $ a $ is the acceleration (which is $ -g $ because the acceleration due to gravity is downward), and $ t $ is the time.

    Therefore, $$ v_{fy} = 12 , \text{m/s} - 10 , \text{m/s}^2 \times 2 , \text{s} $$ Simplifying, $$ v_{fy} = 12 , \text{m/s} - 20 , \text{m/s} = -8 , \text{m/s} $$

  3. Interpret the result: The negative sign indicates the direction of the velocity is downward.

Hence, the vertical component of the velocity after 2 seconds is 8 m/s downward.

Final Answer: B. 8 m/s downward.


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