A tractor of mass 5.0×10^3kg is used to tow a car of mass 2.5×10^3kg. The tractor moved with a speed of 3.0m/s,just before the towing rope becomes taut.calculate:
1. The speed of the tractor immediately after the rope becomes taut.
2. loss in kinetic energy of a system just after the car has started moving.
3. impulse in the rope when it jerks a car into motion.
initial momentum of system = 3*5*10*3 + 0 = 15*10^3
final momentum of system = v * 5*10^3 + v* 2.5 * 10^3 = v* 7.5*10^3
same momentum before and after (Newton's first thought)
v = 15/7.5 = 2
Ke before = (1/2)(5*10^3) * 9
Ke after = (1/2)(7.5*10^3) * 4
loss = (1/2)(10*3)(45 - 30) Joules
impulse = force on car * time = change in car momentum
= 2.5*10^3 * ( 2 - 0)
To answer these questions, we will need to apply the principle of conservation of momentum and kinetic energy.
1. The speed of the tractor immediately after the rope becomes taut:
Before the rope becomes taut, the total initial momentum of the system (tractor + car) is zero because the tractor was stationary. To find the speed of the tractor immediately after the rope becomes taut, we can use the principle of conservation of momentum.
The momentum of an object is given by the product of its mass and velocity. The initial momentum of the system is given by:
Initial momentum = (mass of tractor) × (velocity of tractor before taut rope) + (mass of car) × (velocity of car before taut rope)
Since the tractor was stationary, its initial velocity is 0 m/s. Therefore, the initial momentum of the system is:
Initial momentum = (5.0×10^3 kg) × (0 m/s) + (2.5×10^3 kg) × (3.0 m/s)
After the rope becomes taut, both the tractor and the car will move together. Let's assume the speed of the tractor after the rope becomes taut is v.
The final momentum can then be calculated as:
Final momentum = (mass of tractor) × (velocity of tractor after taut rope) + (mass of car) × (velocity of car after taut rope)
Since both the tractor and the car move together, their velocities are the same, which is v.
Using the principle of conservation of momentum, we can set the initial momentum equal to the final momentum. So, we have:
Initial momentum = Final momentum
(2.5×10^3 kg) × (3.0 m/s) = (5.0×10^3 kg + 2.5×10^3 kg) × v
Now, we can solve for v:
v = (2.5×10^3 kg) × (3.0 m/s) / (5.0×10^3 kg + 2.5×10^3 kg)
Calculate the value and you'll find the speed of the tractor immediately after the rope becomes taut.
2. The loss in kinetic energy of the system just after the car has started moving:
The initial kinetic energy of the system is given by:
Initial kinetic energy = (1/2) × (mass of tractor) × (velocity of tractor before taut rope)^2 + (1/2) × (mass of car) × (velocity of car before taut rope)^2
The final kinetic energy of the system is given by:
Final kinetic energy = (1/2) × (mass of tractor) × (velocity of tractor after taut rope)^2 + (1/2) × (mass of car) × (velocity of car after taut rope)^2
The loss in kinetic energy is the difference between the initial and final kinetic energies:
Loss in kinetic energy = Initial kinetic energy - Final kinetic energy
Substitute the appropriate values into the equations and calculate the loss in kinetic energy.
3. The impulse in the rope when it jerks a car into motion:
Impulse is the change in momentum of an object. In this case, the impulse in the rope will be equal to the change in momentum of the car.
The change in momentum can be calculated as:
Change in momentum = (mass of car) × (velocity of car after taut rope) - (mass of car) × (velocity of car before taut rope)
Substitute the appropriate values into the equation and calculate the impulse in the rope.