Question

pre - and post - collision information is shown. identify the collision parameters that are consistent with the indicated change in momentum.
initial state → collision parameters → final state
v = 4 m/s
3 kg
v = 12 m/s
3 kg
tap an option below to select or change an answer.
(a + sign indicates a rightward direction; a - sign indicates a leftward direction.)
f = +48 n, δt = 3 s
impulse = +16 n·s
f = +24 n, δt = 3 s
impulse = +24 n·s
δp = +48 kg·m/s

Explanation:

Step1: Recall the formula for impulse - momentum theorem

The impulse - momentum theorem states that $J=\Delta p$, where $J$ is the impulse and $\Delta p$ is the change in momentum. Also, impulse $J = F\Delta t$, where $F$ is the force and $\Delta t$ is the time interval.

Step2: Calculate the initial momentum

The initial momentum $p_i=mv_i$, with $m = 3\ kg$ and $v_i=4\ m/s$. So $p_i=3\times4=12\ kg\cdot m/s$.

Step3: Calculate the final momentum

The final momentum $p_f=mv_f$, with $m = 3\ kg$ and $v_f = 12\ m/s$. So $p_f=3\times12 = 36\ kg\cdot m/s$.

Step4: Calculate the change in momentum

$\Delta p=p_f - p_i=36 - 12=24\ kg\cdot m/s$.

Step5: Check the impulse - force relations

For $F = + 24\ N$ and $\Delta t=3\ s$, the impulse $J=F\Delta t=24\times3=72\ N\cdot s$.
For $F = + 48\ N$ and $\Delta t = 3\ s$, the impulse $J=F\Delta t=48\times3 = 144\ N\cdot s$.
The correct impulse value (equal to the change in momentum) is $J=\Delta p=24\ kg\cdot m/s$. When $F = + 24\ N$ and $\Delta t=1\ s$, $J=F\Delta t=24\times1=24\ N\cdot s$.

Answer:

Impulse $=+24\ N\cdot s$