Question

Your friend (68 kg) is wearing frictionless roller skates and standing still. You throw at her a 3.6 kg pumpkin with a velocity of 9.5 m/s and she catches it. What will be her resultant velocity ?

Answer 1

Answer:

Approximately $0.48\; \rm m\cdot s^{-1}$.

Explanation:

Momentum would be conserved since there's no friction on this friend, and all other forces on her are balanced. Therefore:

$\begin{aligned}& \text{Resultant momentum of (friend and pumpkin)} \\ =\; & \text{Initial momentum of (friend)} \\ & + \text{Initial momentum of (pumpkin)}\end{aligned}$.

Momentum $p$ the product of mass $m$ and velocity $v$. That is: $p = m \, v$.

The initial momentum of this friend is $0\; \rm kg \cdot m \cdot s^{-1}$ since she was initially not moving (an initial velocity of $0\; \rm m\cdot s^{-1}$.)

The initial momentum of the pumpkin would be:

$\begin{aligned}p &= m \, v \\ &= 3.6 \; \rm kg \times 9.5\; \rm m\cdot s^{-1} \\ &= 34.2\; \rm kg \cdot m \cdot s^{-1}\end{aligned}$.

Therefore:

$\begin{aligned}& \text{Resultant momentum of (friend and pumpkin)} \\ =\; & \text{Initial momentum of (friend)} \\ & + \text{Initial momentum of (pumpkin)} \\ =\; &0\; {\rm kg \cdot m \cdot s^{-1}} + 34.2\; {\rm kg \cdot m \cdot s^{-1}} \\ =\; & 34.2\; {\rm kg \cdot m \cdot s^{-1}}\end{aligned}$.

Rearrange the equation $p = m \, v$ to find an expression for velocity $v$ given momentum and mass:

$\displaystyle v = \frac{p}{m}$.

Note that the "final momentum of friend and pumpkin" in the previous equation refers to the resultant velocity of the friend with the pumpkin in her hand. Thus, it would necessary to use the combined mass of the friend and the pumpkin $(68\; {\rm kg} + 3.6 \; {\rm kg})$ when calculating the resultant velocity:

$\begin{aligned}& \text{Resultant velocity of (friend and pumpkin)} \\ =\; & \frac{\text{Resultant momentum of (friend and pumpkin)}}{\text{Mass of (friend and pumpkin)}} \\ =\; & \frac{34.2\; {\rm kg \cdot m \cdot s^{-1}}}{68\; {\rm kg} + 3.6\; {\rm kg}} \\ \approx \; & 0.48\; \rm m \cdot s^{-1}\end{aligned}$.