Two identical trucks have mass 5400 kg when empty, and the maximum permissible load for each is 8000 kg. The first truck, carrying 4000 kg, is at rest. The second truck plows into it at 63 km/h, and the pair moves away at 40 km/h. As an expert witness, you’re asked to determine whether the second truck was overloaded. What do you report?
Given:
M1 = 5400 + 4000 = 9400 kg, V1 = 0.
M2 = (5400 + L) kg, V2 = 63km/h.
momentum before = momentum after.
M1*V1 + M2*V2 = M1*V + M2*V.
9400 * 0 + (5400+L)*63 = 9400*40 + (5400+L)*40,
(5400+L)*63 = 376,000 + (5400+L)*40,
(5400+L)*23 = 376,000,
5400+L = 16,348,
L = 10,948 kg = Load of truck #2.
Report: Truck #2 was overloaded.
Well, as an expert witness, I must say that the second truck definitely had too much on its plate...or in this case, on its "bed." Let's break it down.
First, let me convert those speeds to meters per second to keep things consistent. 63 km/h is roughly 17.5 m/s, and 40 km/h is about 11.1 m/s.
Since we have two identical trucks, we can assume that the momentum before the collision is equal to the momentum after the collision. Momentum is calculated by multiplying mass and velocity.
The momentum before the collision is simply the momentum of the first truck, which is 5400 kg (mass of truck) multiplied by 0 (velocity because it's at rest). So it's zero.
Now, after the collision, the momentum is calculated by adding the momentum of both trucks together. The second truck has a mass of 5400 kg (same as the first truck) and a velocity of 11.1 m/s.
So the total momentum after the collision is 5400 kg * 11.1 m/s = 59,940 kg*m/s.
Since momentum is conserved, we can conclude that the momentum before the collision must also be 59,940 kg*m/s. But we already established that the first truck was at rest, so its momentum was zero.
This means that all of the momentum after the collision came from the second truck. And if we use the equation for momentum (mass multiplied by velocity), we can solve for the mass of the second truck.
59,940 kg*m/s = mass of the second truck * 11.1 m/s
Simplifying, we get:
mass of the second truck = 59,940 kg*m/s / 11.1 m/s = 5400 kg
So the mass of the second truck is indeed 5400 kg, the same as the first truck. This tells us that the second truck was not overloaded. It's just that the collision caused both trucks to move away at a different speed together.
In conclusion, there's no need to accuse the poor second truck of carrying a heavier load. It was just a case of an unfortunate collision.
To determine whether the second truck was overloaded, we need to analyze the conservation of momentum.
Let's first convert the given speeds from km/h to m/s:
Initial speed of the second truck (before collision) = 63 km/h = 63 * (1000/3600) m/s = 17.5 m/s
Final speed of the combined trucks (after collision) = 40 km/h = 40 * (1000/3600) m/s = 11.1 m/s
Now, let's calculate the momentum of each truck before and after the collision.
Momentum before the collision:
Truck 1 (at rest):
Mass = 5400 kg
Initial velocity = 0 m/s
Momentum = mass * velocity = 5400 kg * 0 m/s = 0 kg·m/s
Truck 2 (moving):
Mass = 5400 kg
Initial velocity = 17.5 m/s
Momentum = mass * velocity = 5400 kg * 17.5 m/s = 94500 kg·m/s
Total momentum before the collision = Momentum of Truck 1 + Momentum of Truck 2 = 0 kg·m/s + 94500 kg·m/s = 94500 kg·m/s
Momentum after the collision:
Combined mass of both trucks = Mass of Truck 1 + Mass of Truck 2 = 5400 kg + 5400 kg = 10800 kg
Final velocity of the combined trucks = 11.1 m/s
Momentum = Mass * Velocity = 10800 kg * 11.1 m/s = 119880 kg·m/s
Since momentum is conserved in a collision, the total momentum before the collision should be equal to the total momentum after the collision. However, in this case, the momentum after the collision is greater than the momentum before the collision, indicating that the second truck was carrying an excessive load. Therefore, I would report that the second truck was overloaded.
To determine whether the second truck was overloaded, we need to consider the principles of conservation of momentum.
Momentum is the product of an object's mass and its velocity. According to the law of conservation of momentum, the total momentum before and after a collision should be the same, provided no external forces act on the system.
Let's first calculate the initial momentum of each truck. Since the first truck is at rest, its initial momentum is zero since its velocity is zero (p1 = m1 * v1 = 0). The second truck's initial momentum can be calculated by converting the given speed from km/h to m/s:
v2_initial = 63 km/h = (63 * 1000) m/3600 s ≈ 17.5 m/s
p2_initial = m2 * v2_initial ≈ 5400 kg * 17.5 m/s
Now, let's calculate the final momentum of the system (consisting of both trucks) after the collision when they move away together. The final velocity of the trucks can be calculated using the principle of conservation of momentum:
p_final = (m1 + m2) * v_final
Given that the pair moves away at 40 km/h, we convert this to m/s:
v_final = 40 km/h = (40 * 1000) m/3600 s ≈ 11.1 m/s
Substituting the values into the equation:
p2_initial = (m1 + m2) * v_final
Solving for m2:
m2 = (p2_initial - m1 * v_final) / v2_initial
Now we substitute the known values:
m2 ≈ (5400 kg * 17.5 m/s - 4000 kg * 11.1 m/s) / 17.5 m/s
After calculating this expression, you'll find the value of m2, which represents the mass of the second truck. If the value of m2 is less than or equal to 8000 kg (the maximum permissible load for each truck), then the second truck was not overloaded. However, if m2 is greater than 8000 kg, then the second truck was overloaded.
So, as an expert witness, you report whether the value of m2 is within or exceeds the maximum permissible load for the second truck to determine if it was overloaded or not.