Answer :
Final answer:
To find the initial height from which a crossbow bolt was shot, we use projectile motion principles. The bolt, fired horizontally, falls 0.21 m in the time it takes to reach the target. Using the displacement formula and the time calculated, the initial height is found to be approximately 0.21 m.
Explanation:
The student's question pertains to the initial height from which a crossbow bolt was fired, given it was shot horizontally and landed 0.21 m below the launch height after traveling 38.1 m to the target. This is a classic physics problem involving projectile motion, where only the vertical motion is affected by gravity since air resistance is being ignored.
To find the initial height, we can use the formula for the vertical displacement in projectile motion, which is Δy = V_i_y × t + 0.5 × g × t^2. Because the crossbow bolt is fired horizontally, the initial vertical velocity V_i_y is 0, simplifying the equation to Δy = 0.5 × g × t^2. Given that Δy = -0.21 m (downward displacement is negative) and g = 9.8 m/s^2 (acceleration due to gravity), we can solve for t.
Using the formula and rearranging for t gives t = √(2×Δy / g). Plugging in the values gives us t = √(2×(-0.21 m) / 9.8 m/s^2) which calculates to approximately t = 0.207 s. Now that we have the time, we can use it to calculate the initial height from which the bolt was fired using the same vertical displacement formula, treating Δy as the initial height h and knowing that it takes t = 0.207 s for the bolt to drop 0.21 m under gravity. Therefore, h = 0.5 × g × t^2, which calculates to h = 0.5 × 9.8 m/s^2 × (0.207 s)^2, giving us an initial height of approximately 0.21 m.
