A truck of mass 2000kg travelling due south collides at an intersection with a car of mass 1200kg travelling due east. The car and truck become entangled and move off to the south-east. The truck driver claims to have been travelling at exactly 60km/hr(the speed limit) just prior to impact. Determine whether the car was exceeding the speed limit.
To have a final southeast direction, the initial momenta of the truck and car must have been equal in magnitude.
If the heavier truck was travelling at the 60 km/h speed limit, the car must have been going 100 km/h.
(100*1200) = (60*2000)
To determine whether the car was exceeding the speed limit, we need to calculate the final velocity of the car and compare it to the speed limit of 60 km/hr.
To solve this problem, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
The momentum of an object is given by the product of its mass and velocity: momentum = mass x velocity.
Before the collision, the momentum of the truck is given by: momentum(truck) = mass(truck) x velocity(truck).
Given that the mass of the truck is 2000 kg and the driver claims to have been traveling at exactly 60 km/hr, we need to convert the velocity from km/hr to m/s.
1 km/hr = (1/3.6) m/s (conversion factor)
So, velocity(truck) = 60 km/hr x (1/3.6) m/s = 16.67 m/s.
Therefore, momentum(truck) before the collision = 2000 kg x 16.67 m/s.
Before the collision, the momentum of the car is given by: momentum(car) = mass(car) x velocity(car).
Given that the mass of the car is 1200 kg, we need to find the velocity(car).
Let's assume the final velocity of the car and truck after the collision is v. Since the car and truck move off in the south-east direction, we can break down this velocity into its north-south and east-west components.
The north-south component of the final velocity is v_car_ns = v x sin(45°), and the east-west component of the final velocity is v_car_ew = v x cos(45°).
According to the principle of conservation of momentum, the total momentum before the collision should be equal to the total momentum after the collision. This can be represented mathematically as:
momentum(car) + momentum(truck) = momentum(final)
m(car) x v_car + m(truck) x v_truck = (m(car) + m(truck)) x v_final
1200 kg x v_car + 2000 kg x 16.67 m/s = (1200 kg + 2000 kg) x v
Simplifying the equation, we can solve for v_car:
1200 kg x v_car + 33340 kg m/s = 3200 kg x v
1200 kg x v_car = 3200 kg x v - 33340 kg m/s
v_car = (3200 kg x v - 33340 kg m/s) / 1200 kg
Now, we can substitute this equation for v_car into the equation for the north-south component of the final velocity:
v_car_ns = [(3200 kg x v - 33340 kg m/s) / 1200 kg] x sin(45°)
Finally, we can substitute the known values into the equation and calculate the north-south component of the final velocity, v_car_ns.
Once we have v_car_ns, we can calculate the magnitude of the final velocity, v_final:
v_final = √[v_car_ns^2 + v_car_ew^2]
If v_final is greater than 60 km/hr, then the car was exceeding the speed limit.