Collision at an Intersection

A $1,550kg$ car traveling east with a speed of $25\mathrm{.}6\frac{m}{s}$ collides at an intersection with a $2,550$ kg truck traveling north at a speed of $22\mathrm{.}8\frac{m}{s}$ as shown in the figure. Find the direction and magnitude of the velocity of the wreckage after the collision, assuming the vehicles stick together after the collision.

SOLUTION

Conceptualize The figure should help you conceptualize the situation before and after the collision. Let us choose east to be along the positive $x-$direction and north to be along the positive $y-$direction.

In what direction is the blue car moving initially?

to the north

to the south

to the east

to the west

Categorize Because we consider moments immediately before and immediately after the collision as defining our time interval, we ignore the small effects that friction would have on the wheels of the vehicles and model the two vehicles as an isolated system in terms of momentum. We also ignore the vehicles' sizes and model them as particles. The collision is perfectly because the car and the truck stick together after the collision.

Analyze Before the collision, the only object having momentum in the $x-$direction is the car hat$(v).$ Therefore, the magnitude of the total initial momentum of the system $($car plus truck$)$ in the $x-$direction is that of only the car. Similarly, the total initial momentum of the system in the $y-$direction is that of the $.$ After the collision, let us assume the wreckage moves at an angle $$ with respect to the $x-$axis with speed $v_f$

$($For the next three answers, use the following as necessary: $m_1,m_2,v_f$ and $.$ Do not substitute numerical values; use variables only.$)$

Apply the isolated system model for momentum in the $x-$direction:

$p_x=0{\displaystyle \underset{?}{\overset{?}{}}}{p}_{}={\displaystyle \underset{?}{\overset{?}{}}}{p}_{xf}$

$(1)m_1{v}_{1i}=(m_1+m_2)$

Apply the isolated system model for momentum in the $y-$direction:

$p_y=0{\displaystyle \underset{?}{\overset{?}{}}}{p}_{yi}={\displaystyle \underset{?}{\overset{?}{}}}{p}_{yf}$

$(2)m_2{v}_{2i}=1$

Divide Equation $(2)$ by Equation $(1)$:

$\frac{m_2{v}_{2i}}{m_1{v}_{1i}}=$

Solve for $($in degrees counterclockwise from the $+x-$axis$)$ and substitute numerical values:

$=ta{n}^{-1}(\frac{m_2{v}_{2i}}{m_1{v}_{1i}})=,$@ counterclockwise from the $+x-$axis

Use Equation $(2)$ to find the value of $v_f$ and substitute numerical values $($Enter the magnitude in $\frac{m}{s}.)$:

Finalize Notice that the final speed of the combination is less than the initial speeds of the two cars. This result is consistent with the kinetic energy of the system being reduced in an inelastic collision. It might help if you draw the momentum vectors of each vehicle before the collision and the two vehicles together after the collision.

EXERCISE

The eastbound car in the example is driven by an off$-$duty police officer, who knows her speed was exactly $30\mathrm{.}0\frac{m}{s}.$ From studying the angle $($and the mass of the two vehicles$),$ she concluded that the truck was over the speed limit of $70k\frac{m}{h}$ driving on that secondary road. Of course, the truck also failed to stop before entering the main road. If the truck were not speeding, what should be the maximum angle for $($in degrees counterclockwise from the $+x-$axis$)$ in this accident?

Hint

$$

$$ counterclockwise from the $+x-$axis