Question

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 -

Answer 1

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.