A car with a mass of 980 kg and a speed of v1 = 17.0 m/s approaches an intersection, as shown in the figure(Figure 1) . A 1300 kg minivan traveling at v2 is heading for the same intersection. The car and minivan collide and stick together. The direction of the wreckage after the collision is θ = 41.5 ∘ above the x axis.
Find the initial speed of the minivan, assuming external forces can be ignored.
Part B
Find the final speed of the wreckage.
To find the initial speed of the minivan (v2) and the final speed of the wreckage, we can use the principle of conservation of linear momentum.
The conservation of linear momentum states that the total momentum of an isolated system remains constant before and after a collision, assuming no external forces act on the system. Mathematically, this can be expressed as:
\(m1 * v1 + m2 * v2 = (m1 + m2) * vf\)
where:
m1 and m2 are the masses of the car and minivan respectively,
v1 and v2 are the initial velocities of the car and minivan respectively,
vf is the final velocity of the wreckage.
In this case, we are given the mass of the car (m1 = 980 kg), the speed of the car (v1 = 17.0 m/s), the mass of the minivan (m2 = 1300 kg), and the angle θ = 41.5° above the x-axis.
Step 1: Find the initial velocity of the minivan (v2)
To solve for v2, we rearrange the equation as follows:
\(m1 * v1 = (m1 + m2) * vf - m2 * v2\)
\(v2 = \frac{{m1 * v1 + m2 * vf - m1 * vf}}{{m2}}\)
Plugging in the given values, we have:
\(v2 = \frac{{(980 kg * 17.0 m/s) + (1300 kg * vf) - (980 kg * vf)}}{{1300 kg}}\)
Step 2: Find the final velocity of the wreckage (vf)
To solve for vf, we can use the momentum conservation equation once again:
\(m1 * v1 + m2 * v2 = (m1 + m2) * vf\)
Plugging in the given values, including the initial velocity we found in step 1, we have:
\(980 kg * 17.0 m/s + 1300 kg * v2 = (980 kg + 1300 kg) * vf\)
Now we have two equations with two unknowns (v2 and vf), which can be solved simultaneously.
Once we solve the equations, we will have the initial velocity of the minivan (v2) and the final velocity of the wreckage (vf).