Question

A car weighs 3600 kg is traveling at 21. 0 m/s. The driver doesn?t notice a red light and rear-ends another car at rest. Both cars stick together and move with a speed of 9. 0 m/s. What is the mass of the car that was at rest?.

Answer 1

`4800\; {\rm kg}`.

Step-by-step explanation:

When an object of mass
`m` travels at a velocity of
`v`, the momentum
`p` of that object would be
`p = m\, v`.

Momentum is conserved immediately after the collision. Hence, the momentum that the
`3600\; {\rm kg}` vehicle lost in the collision would be equal to the momentum that the other vehicle has gained.

Momentum that the
`3600\; {\rm kg}` vehicle lost:

`\begin{aligned}(\text{momentum change}) &= (\text{new momentum}) - (\text{original momentum}) \\ &= m\, v(\text{new}) - m\, v(\text{old}) \\ &= m\, (v(\text{new}) - v(\text{old})) \\ &= 3600\; {\rm kg} \, (9.0\; {\rm m\cdot s^(-1)} - 21.0\; {\rm m\cdot s^(-1)}) \\ &= (-43200)\; {\rm kg \cdot m \cdot s^(-1)}\end{aligned}`.

The vehicle that was originally at rest would have gained that
`43200\; {\rm kg \cdot m\cdot s^(-1)}` of momentum. The momentum of that vehicle was
`0\; {\rm kg \cdot m \cdot s^(-1)}` before the collision since that vehicle was initially not moving. After gaining the
`43200\; {\rm kg \cdot m\cdot s^(-1)}\!` of momentum, the new momentum of that vehicle would be
`p = 43200\; {\rm kg \cdot m\cdot s^(-1)}\!\!`.

Rearrange the equation
`p = m\, v` to obtain an expression for mass:
`m = (p / v)`. It is given that the velocity of this vehicle is
`v = 9.0\; {\rm m\cdot s^(-1)}` after the collision. Substitute both momentum
`p` and velocity
`v` into the equation
`m = (p / v)\!` to find mass
`m\!`:

`\begin{aligned} p &= (m)/(v) \\ &= \frac{43200\; {\rm kg \cdot m \cdot s^(-1)}}{9.0\; {\rm m\cdot s^(-1)}} \\ &= 4800\; {\rm kg} \end{aligned}`.

In other words, the mass of the vehicle that was originally not moving would be
`4800\; {\rm kg}`.