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Plsss help Bumper car A (282 kg) moving +2.82 m/s

makes an elastic collision with bumper
car B (210 kg) moving +1.72 m/s. What is
the velocity of car A after the collision?
(Unit = m/s)
Remember: right is +, left is -

Plsss help Bumper car A (282 kg) moving +2.82 m/s makes an elastic collision with-example-1
User Despatcher
by
7.7k points

1 Answer

4 votes

Answer:

Approximately
1.89\; {\rm m\cdot s^(-1)}.

Step-by-step explanation:

Let
m_(A) and
m_(B) denote the mass of the two vehicles. Let
u_(A) and
u_(B) denote the velocity before the collision. Let
v_(A) and
v_(B) denote the velocity after the collision.

Since the collision is elastic, both momentum and kinetic energy should be conserved.

For momentum to conserve:


m_(A) \, v_(A) + m_(B) \, v_(B) = m_(A)\, u_(A) + m_(B)\, u_(B).

For kinetic energy to conserve:


\displaystyle (1)/(2)\, m_(A) \, ({v_(A)}^(2)) + (1)/(2)\, m_(B) \, ({v_(B)}^(2)) = (1)/(2)\, m_(A)\, ({u_(A)}^(2)) + (1)/(2)\, m_(B)\, ({u_(B)}^(2)).

Simplify to obtain:


\displaystyle m_(A) \, ({v_(A)}^(2)) + m_(B) \, ({v_(B)}^(2)) = m_(A)\, ({u_(A)}^(2)) + m_(B)\, ({u_(B)}^(2)).

It is given that
m_(A) = 282\; {\rm kg},
m_(B) = 210\; {\rm kg},
u_(A) = 2.82\; {\rm m\cdot s^(-1)}, and
u_(B) = 1.72\; {\rm m\cdot s^(-1)}. The value (in
{\rm m\cdot s^(-1)}) of
v_(A) and
v_(B) can be found by solving this nonlinear system of two equations and two unknowns:


\left\lbrace \begin{aligned} & m_(A) \, v_(A) + m_(B) \, v_(B) = m_(A)\, u_(A) + m_(B)\, u_(B) \\ & m_(A) \, ({v_(A)}^(2)) + m_(B) \, ({v_(B)}^(2)) = m_(A)\, ({u_(A)}^(2)) + m_(B)\, ({u_(B)}^(2))\end{aligned}\right..


\left\lbrace \begin{aligned} & 282 \, v_(A) + 210 \, v_(B) = 282\, (2.82) + 210\, (1.72) \\ & 282 \, ({v_(A)}^(2)) + 210 \, ({v_(B)}^(2)) = 282\, ({2.82}^(2)) + 210\, ({1.72}^(2))\end{aligned}\right..

Solving this system gives two possible sets of solutions:


  • \left\lbrace\begin{aligned}v_(A) &\approx 1.89\; {\rm m\cdot s^(-1)} \\ v_(B) &\approx 2.98\; {\rm m\cdot s^(-1)}\end{aligned}\right..

  • \left\lbrace\begin{aligned}v_(A) &\approx 2.82\; {\rm m\cdot s^(-1)} \\ v_(B) &\approx 1.72\; {\rm m\cdot s^(-1)}\end{aligned}\right..

However, the second set of solutions is invalid since it suggests that the velocity of the two vehicles stayed unchanged after the collision. Hence, only the first set of solutions (
v_(A) &\approx 1.89\; {\rm m\cdot s^(-1)},
v_(B) &\approx 2.98\; {\rm m\cdot s^(-1)}) is valid.

Therefore, the velocity of vehicle
A would be approximately
1.89\; {\rm m\cdot s^(-1)} after the collision.

User UncleIstvan
by
8.0k points
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