During winter break, the police department in your hometown asks you to consult in the investigation of a traffic accident that occurred in a large parking lot. It seems that a car was backing out of a parking space when a pickup truck plowed straight into it from behind. The driver of the pickup truck claims that he was going 10 mph but police would like you to perform an analysis to verify this. The car was moving 5 mph as it backed up in a straight-line path (three witnesses inside the car independently gave that information so police are trusting it for now). When the two vehicles collided, both drivers immediately slammed on the brakes to lock the wheels. The bumpers crumpled and the two vehicles stuck together. Skid marks show that they traveled 5 meters forward in a straight line before coming to rest. The police have already performed some measurements: the mass of the car is 1500 kg, the mass of the truck is 2500 kg, and the coefficient of kinetic friction between the tire rubber and the asphalt of the parking lot is 0.49
there's no question, just a bunch of information about the situation
They want you to decide whether or not the truck driver was actually going 1o mph or not.
What I have so far is v2=(m1v1)/-m2
or v2=(1500kg x 16.093 km/hr)/-2500 kg
which gives -9.6558
However, that seems to simplistic so it's probably super wrong.
m₁=1500 kg, m₂=2500 kg,
v₁=5mph=2.24 m/s, claimed velocity is u=10 mph=4.47 m/s,
real speed of the pickup is v₂=?
μ=0.49, s=5 m.
Law of conservation of linear momentum
m₁•v₁+m₂•v₂=(m₁+m₂)•v,
law of conservation of energy
KE -> W(fr)
(m₁+m₂)•v²/2= μ•(m₁+m₂)•g•s.
v=sqrt(2•μ•g•s)=sqrt(2•0.49•9.8•5)=6.93 m/s.
v₂={(m₁+m₂)•v - m₁•v₁}/m₂=
={(1500+2500)•6.93 -1500•2.24}/2500 =9.74 m/s=24.8 mph
24.8 mph > 10 mph
To verify the driver's claim that the pickup truck was moving at 10 mph, we need to perform an analysis using the given measurements and information.
First, we need to find the initial velocities of both vehicles before the collision. We know that the car was moving at 5 mph (miles per hour), so we'll convert it to meters per second (m/s) for consistent units.
1 mile = 1609.34 meters
1 hour = 3600 seconds
Therefore, the initial velocity of the car is:
v_car = (5 mph) * (1609.34 m/1 mile) * (1 hour/3600 seconds) ≈ 2.235 m/s
Next, we'll use the concept of conservation of momentum to analyze the collision. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
The momentum (p) of an object is calculated by multiplying its mass (m) by its velocity (v):
p = m * v
Let's denote the initial velocity of the truck as v_truck.
The total momentum of the system before the collision is given by:
p_initial = p_car + p_truck
Since both drivers immediately locked their wheels and the vehicles stuck together after the collision, we can assume the collision was perfectly inelastic. Hence, the two vehicles moved together as one unit after the collision.
The final velocity of the combined vehicles (car + truck) is zero because they came to rest. Therefore, the total momentum after the collision is:
p_final = (m_car + m_truck) * 0
Using the given masses of the car (1500 kg) and the truck (2500 kg), we have:
p_final = (1500 + 2500) kg * 0 m/s = 0 kg * m/s
Now, equating the initial and final momenta:
p_initial = p_final
(m_car * v_car) + (m_truck * v_truck) = 0
Substituting the known values:
(1500 kg * 2.235 m/s) + (2500 kg * v_truck) = 0
Now, solve for v_truck:
2500 kg * v_truck = -1500 kg * 2.235 m/s
v_truck = (-1500 kg * 2.235 m/s) / 2500 kg
v_truck ≈ -1.34 m/s
However, since speed is defined as a scalar quantity, it cannot be negative. Therefore, we take the magnitude of the velocity to calculate the speed:
speed_truck = |-1.34 m/s| = 1.34 m/s
Finally, to convert the speed from meters per second (m/s) to miles per hour (mph) for comparison with the driver's claim:
speed_truck = 1.34 m/s * (3600 seconds/1 hour) * (1 mile/1609.34 m)
speed_truck ≈ 2.99 mph
According to the analysis, the speed of the pickup truck was approximately 2.99 mph, which is significantly lower than the driver's claim of 10 mph. Therefore, the analysis contradicts the driver's statement, suggesting that the truck was not going 10 mph.
Note: The negative sign of the velocity indicates that the direction of the truck's motion was opposite to the forward direction of the car.