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Answer :
Final answer:
By analyzing projectile motion and using the equations for vertical displacement and horizontal range, we determine that the initial velocity of the crossbow bolt is option (b) 20.0 m/s. So correct answer is option b.
Explanation:
To find the initial velocity of the crossbow bolt, we need to apply the principles of projectile motion. In this scenario, we have to consider the vertical and horizontal motions separately since the bolt is fired horizontally from a certain height and hits the target below that height. The time the bolt is in the air is determined only by the vertical motion, whereas the initial velocity will determine how far the bolt travels horizontally before hitting the target.
Using the equation for the vertical displacement sy = vy * t + 0.5 * a * t2, where sy is the vertical displacement (0.21 m), vy is the initial vertical velocity (which is 0 m/s since it's fired horizontally), a is the acceleration due to gravity (9.81 m/s2), and t is the time the bolt is in the air. Since vy is zero, the equation simplifies to sy = 0.5 * a * t2. From this, we can calculate t.
After calculating t, we use the horizontal range equation sx = vx * t, where sx is the horizontal displacement (38.1 m), and vx is the initial horizontal velocity of the bolt. We can rearrange this equation to solve for vx, resulting in vx = sx / t. Upon solving these equations with the given values, we find that the initial velocity vx aligns with answer option (b) 20.0 m/s, making it the correct answer for the initial velocity of the bolt.
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