Question

The greatest speed with which an athlete can jump vertically is around 5 m/sec. Determine the speed at which Earth would move down if a 664 N athelete jumped up at 1.8 m/sec. (Earth's mass = 5.97 ✕ 1024 kg)

Answer 1

Approximately
`2.0 * 10^(-23)\; \rm m \cdot s^(-1)` if that athlete jumped up at
`1.8\; \rm m \cdot s^(-1)`. (Assuming that
`g = 9.81\; \rm m\cdot s^(-1)`.)

Step-by-step explanation:

The momentum
`p` of an object is the product of its mass
`m` and its velocity
`v`. That is:
`p = m \cdot v`.

Before the jump, the speed of the athlete and the earth would be zero (relative to each other.) That is:
`v(\text{athlete, before}) = 0` and
`v(\text{earth, before}) = 0`. Therefore:

`\begin{aligned}& p(\text{athlete, before}) = 0\end{aligned}` and
`p(\text{earth, before}) = 0`.

Assume that there is no force from outside of the earth (and the athlete) acting on the two. Momentum should be conserved at the instant that the athlete jumped up from the earth.

Before the jump, the sum of the momentum of the athlete and the earth was zero. Because momentum is conserved, the sum of the momentum of the two objects after the jump should also be zero. That is:

`\begin{aligned}& p(\text{athlete, after}) + p(\text{earth, after}) \\ & =p(\text{athlete, before}) + p(\text{earth, before}) \\ &= 0\end{aligned}`.

Therefore:

`p(\text{athelete, after}) = - p(\text{earth, after})`.

`\begin{aligned}& m(\text{athlete}) \cdot v(\text{athelete, after}) \\ &= - m(\text{earth}) \cdot v(\text{earth, after})\end{aligned}`.

Rewrite this equation to find an expression for
`v(\text{earth, after})`, the speed of the earth after the jump:

`\begin{aligned} &v(\text{earth, after}) \\ &= -\frac{m(\text{athlete}) \cdot v(\text{athlete, after})}{m(\text{earth})} \end{aligned}`.

The mass of the athlete needs to be calculated from the weight of this athlete. Assume that the gravitational field strength is
`g = 9.81\; \rm N \cdot kg^(-1)`.

`\begin{aligned}& m(\text{athlete}) = (664\; \rm N)/(9.81\; \rm N \cdot kg^(-1)) \approx 67.686\; \rm N\end{aligned}`.

Calculate
`v(\text{earth, after})` using
`m(\text{earth})` and
`v(\text{athlete, after})` values from the question:

`\begin{aligned} &v(\text{earth, after}) \\ &= -\frac{m(\text{athlete}) \cdot v(\text{athlete, after})}{m(\text{earth})} \\ &\approx -2.0 * 10^(-23)\; \rm m \cdot s^(-1)\end{aligned}`.

The negative sign suggests that the earth would move downwards after the jump. The speed of the motion would be approximately
`2.0 * 10^(-23)\; \rm m \cdot s^(-1)`.