Question

question 1
you are trying to escape a hoard of zombies and must make an elevator to get down a cliff as fast as possible. as you cannot hit the ground faster than 10m/s and the cliff is 25m deep, this limits your acceleration to 2 m/s² (you will want to accelerate at this rate to get down the cliff as fast as you can). assuming your and the buckets mass together is 100kg (m2), what is the minimum mass (m1) that the sled must be to prevent your death (assume the ground is frictionless).

Explanation:

Step1: Apply Newton's second - law

Let the acceleration of the system be $a = 2m/s^{2}$, $m_2=100kg$, and the tension in the rope be $T$. For the mass $m_2$ (you and the bucket), according to Newton's second - law $m_2g - T=m_2a$. So, $T = m_2(g - a)$.
Substitute $g = 10m/s^{2}$, $m_2 = 100kg$, and $a = 2m/s^{2}$ into the formula:
$T=100\times(10 - 2)=800N$.

Step2: Analyze the motion of $m_1$

For the mass $m_1$ on the frictionless surface, the force acting on it is the tension $T$. According to Newton's second - law $T = m_1a$.
We know $T = 800N$ and $a = 2m/s^{2}$, then $m_1=\frac{T}{a}$.
Substitute $T = 800N$ and $a = 2m/s^{2}$ into the formula:
$m_1=\frac{800}{2}=400kg$.

Answer:

400