a truck of mass 5000kg is used to tow a jeep of mass 2500kg, the truck moves with a speed of 3.0ms(power -1) just before the towing rope becomes taut,calculate the speed of the truck just before the towing rope became taut,loss of kinetic energy of the system just after the Jeep has started moving,and the impulse in the rope when it jerks the Jeep into motion
To calculate the speed of the truck just before the towing rope becomes taut, we can use the principle of conservation of momentum. Before the Jeep starts moving, the momentum of the system is zero since both the truck and the Jeep are at rest. When the Jeep starts moving, it will acquire momentum in the opposite direction to the motion of the truck. We can set up the following equation:
(mass of truck) * (initial velocity of truck) = (mass of Jeep) * (velocity of Jeep)
Given:
Mass of truck (m1) = 5000 kg
Mass of Jeep (m2) = 2500 kg
Initial velocity of truck (u1) = 3.0 m/s
Using the equation: m1 * u1 = m2 * v2
Rearranging and substituting the values gives us:
(5000 kg) * (3.0 m/s) = (2500 kg) * v2
15,000 kg⋅m/s = 2,500 kg * v2
v2 = (15,000 kg⋅m/s) / (2,500 kg) = 6.0 m/s
Therefore, the speed of the truck just before the towing rope becomes taut is 6.0 m/s.
Now, to calculate the loss of kinetic energy of the system just after the Jeep has started moving, we can use the equation for kinetic energy:
Kinetic energy = 1/2 * mass * velocity^2
Initially, the kinetic energy of the system is zero, as both the truck and the Jeep are at rest. After the Jeep starts moving, it gains kinetic energy while the truck loses some of its initial kinetic energy.
The loss of kinetic energy of the system can be calculated as the difference between the initial kinetic energy and the final kinetic energy of the truck. The final kinetic energy of the truck can be found using the mass of the truck and its final velocity.
Initial kinetic energy of the truck (KE_initial) = 1/2 * mass of truck * (initial velocity of truck)^2
Final kinetic energy of the truck (KE_final) = 1/2 * mass of truck * (final velocity of truck)^2
Loss of kinetic energy = KE_initial - KE_final
Given:
Mass of truck = 5000 kg
Initial velocity of truck = 3.0 m/s
Final velocity of truck = 6.0 m/s
KE_initial = 1/2 * (5000 kg) * (3.0 m/s)^2 = 22,500 J
KE_final = 1/2 * (5000 kg) * (6.0 m/s)^2 = 90,000 J
Loss of kinetic energy = 22,500 J - 90,000 J = -67,500 J
Therefore, the loss of kinetic energy of the system just after the Jeep has started moving is -67,500 J.
Lastly, to calculate the impulse in the rope when it jerks the Jeep into motion, we can use the principle of impulse-momentum.
Impulse = change in momentum
The change in momentum can be calculated by finding the difference between the final momentum (mass of Jeep * velocity of Jeep) and the initial momentum (zero since the Jeep was initially at rest).
Impulse = (mass of Jeep) * (velocity of Jeep) - 0
Given:
Mass of Jeep = 2500 kg
Velocity of Jeep (v2) = 6.0 m/s (as calculated earlier)
Impulse = (2500 kg) * (6.0 m/s) = 15,000 kg⋅m/s
Therefore, the impulse in the rope when it jerks the Jeep into motion is 15,000 kg⋅m/s.