Question

please assume the free-fall acceleration \(g=9.80 \mathrm{m} / \mathrm{s}^{2}\) unless a more precise value is given in the problem statement. Ignore air resistance. Superman is standing \(120 \mathrm{m}\) horizontally away from Lois Lane. A villain throws a rock vertically downward with a speed of $2.8 \mathrm{m} / \mathrm{s}\( from \)14.0 \mathrm{m}$ directly above Lois. (a) If Superman is to intervene and catch the rock just before it hits Lois, what should be his minimum constant acceleration? (b) How fast will Superman be traveling when he reaches Lois?

Short answer

Answer: The minimum constant acceleration Superman needs is approximately 23.56 m/s², and his speed when he reaches Lois is approximately 41.96 m/s.

Step-by-step solution

Calculate the time it takes for the rock to fall

First, we need to find out how long it takes for the rock to fall from 14.0 m above Lois Lane. We can use this equation of motion: $$ y_{final} = y_{initial} + v_{initial} * t - 0.5 * g * t^{2} $$ Using the given information, we have: $$ 0 \mathrm{m} = 14.0 \mathrm{m} + 2.8 \mathrm{m / s} * t - 0.5 * 9.80 \mathrm{m / s^{2}} * t^{2} $$ Solving for t, we get: $$ t \approx 1.781 \mathrm{s} $$ It takes approximately 1.781 seconds for the rock to reach Lois Lane.

Calculate the minimum constant acceleration for Superman

Now, we need to find the minimum constant acceleration Superman needs to catch the rock before it reaches Lois. Using the equation of motion: $$ x_{final} = x_{initial} + v_{initial} * t + 0.5 * a * t^{2} $$ According to the problem statement, Superman is standing 120 m horizontally away from Lois. His initial position is 0 m, and his final position should be 120 m. Superman's initial velocity is 0 m/s since he starts from rest. We want to find a, the minimum constant acceleration, and we know the time t should be less than or equal to the time it takes for the rock to fall (1.781 s). Plugging in the values, we get: $$ 120 \mathrm{m} = 0 \mathrm{m} + 0 \mathrm{m / s} * t + 0.5 * a * (1.781 \mathrm{s})^{2} $$ Solving for a, we get: $$ a \approx 23.56 \mathrm{m / s^{2}} $$ Superman's minimum constant acceleration should be approximately 23.56 m/s².

Calculate Superman's speed when reaching Lois

Finally, we need to find Superman's speed when he reaches Lois. We can use the equation of motion: $$ v_{final} = v_{initial} + a * t $$ Superman's initial velocity is 0 m/s, and we found his constant acceleration to be approximately 23.56 m/s². Using the time it takes for the rock to fall (1.781 s), we can find his final speed: $$ v_{final} = 0 \mathrm{m / s} + 23.56 \mathrm{m / s^{2}} * 1.781 \mathrm{s} $$ Solving for \(v_{final}\), we get: $$ v_{final} \approx 41.96 \mathrm{m / s} $$ Superman's speed when he reaches Lois is approximately 41.96 m/s.