1) What average power is required for a runner with a mass of 85 kg to reach a speed of 8 m/s from
a stationary start during a time of 2 s?
2) A 1000 kg car travels east at 30 m/s and collides with a 3400 kg truck travelling south at 20 m/s.
After the collision the two vechicles remain stuck together.
(a) What is the total initial momentum of the car truck system?
(b) What is the total final momentum of the car truck system?
(c) In what direction are the car and truck moving after the collision?
(d) Immediately after the collision how fast are the car and truck moving?
(e) How much kinetic energy is lost in this collision? Where does this energy go?
1. P = Mg * Vavg = 85*9.8 * 8/2 = 3332 J/s = 3332 Watts
2a. Momentum = M1*V1 + M2*V2 =
1000*30 + 3400*20i = 30,000 - 68,000i,Q4
= sqrt(30,000^2+68000^2) = 74,324 kg-m/s
2b. Total final momentum = Total initial
momentum = 74,324 kg-m/s.
2c. M1*V1 + M2*V2 = M1*V + M2*V
30,000 - 68000i=1000*V + 3400*V = 4400V
4400V = 74324[-66.2o]
V = 16.9 m/s[-66.2o] = 16.9 m/s at 66.2o
S. of E.
2d. V = 16.9 m/s(Part 2c).
1) To find the average power required for the runner, we can use the equation:
Average power = Work / Time
First, let's calculate the work done by the runner to reach the given speed. The work done is equal to the change in kinetic energy.
Change in kinetic energy = 0.5 * mass * (final velocity^2 - initial velocity^2)
Given:
Mass (m) = 85 kg
Initial velocity (u) = 0 m/s (stationary start)
Final velocity (v) = 8 m/s
Time taken (t) = 2 s
Plugging in the values:
Change in kinetic energy = 0.5 * 85 kg * (8 m/s)^2 - 0.5 * 85 kg * (0 m/s)^2
= 0.5 * 85 kg * (64 m/s^2) - 0.5 * 85 kg * 0 m/s^2
= 0.5 * 85 kg * 64 m^2/s^2
= 2176 J
Now, let's calculate the average power:
Average power = 2176 J / 2 s
= 1088 W
Therefore, the average power required for the runner to reach a speed of 8 m/s in 2 seconds is 1088 watts.
2) (a) The total initial momentum of the car-truck system can be calculated by adding the individual momenta of the car and truck:
Initial momentum of car = mass of car * velocity of car
= 1000 kg * 30 m/s
= 30000 kg·m/s
Initial momentum of truck = mass of truck * velocity of truck
= 3400 kg * 20 m/s
= 68000 kg·m/s
Total initial momentum of the car-truck system = Initial momentum of car + Initial momentum of truck
= 30000 kg·m/s + 68000 kg·m/s
= 98000 kg·m/s
(b) Since the car and truck stick together after the collision, the total final momentum of the car-truck system is equal to the total initial momentum:
Total final momentum of the car-truck system = 98000 kg·m/s
(c) To determine the direction in which the car and truck are moving after the collision, we need to consider the vectors. The car was initially moving east and the truck was initially moving south. The final vector sum of their velocities determines the direction.
(d) To find the speed of the car and truck immediately after the collision, we divide the total final momentum by the total mass:
Total mass of the car-truck system = mass of car + mass of truck
= 1000 kg + 3400 kg
= 4400 kg
Car and truck speed immediately after the collision = Total final momentum / Total mass
= 98000 kg·m/s / 4400 kg
= 22.27 m/s
(e) The kinetic energy lost in the collision can be calculated by finding the difference between the initial kinetic energy and the final kinetic energy of the car-truck system.
Initial kinetic energy = 0.5 * mass of car * (velocity of car)^2
= 0.5 * 1000 kg * (30 m/s)^2
= 450000 J
Final kinetic energy = 0.5 * total mass of the car-truck system * (speed after the collision)^2
= 0.5 * 4400 kg * (22.27 m/s)^2
= 1091264 J
Kinetic energy lost in the collision = Initial kinetic energy - Final kinetic energy
= 450000 J - 1091264 J
= -641264 J
The negative sign indicates that energy was lost during the collision. This lost kinetic energy is transformed into other forms, such as heat, sound, or deformation of the vehicles.
To answer the first question, we can use the equation for power: power = work/time. In this case, we need to find the work done on the runner to reach the given speed. The work done on an object can be calculated using the equation work = change in kinetic energy.
To find the change in kinetic energy, we use the equation for kinetic energy: kinetic energy = (1/2) * mass * velocity^2. Since the runner starts from rest, the initial kinetic energy is 0. The final kinetic energy can be calculated using the given mass and final velocity.
Once we have the change in kinetic energy, we can substitute it along with the given time into the power equation to find the average power required by the runner.
For the second question, we can use the conservation of momentum principle, which states that the total momentum before a collision is equal to the total momentum after the collision.
(a) To find the total initial momentum of the car-truck system, we multiply the mass of the car by its velocity in the east direction and the mass of the truck by its velocity in the south direction. The directions are important to consider as momentum is a vector quantity.
(b) After the collision, when the two vehicles remain stuck together, their velocities will have changed. To find the total final momentum of the car-truck system, we again multiply the combined mass by the new velocity.
(c) To determine the direction in which the car and truck are moving after the collision, we can consider the signs of the velocities. Since the car was initially moving east and the truck was moving south, we can determine the resultant direction using vector addition.
(d) The speed of the car and truck immediately after the collision can be found by dividing the total final momentum by the combined mass.
(e) The kinetic energy lost in this collision can be calculated by finding the difference between the initial and final kinetic energies of the car and truck. The energy lost is often converted into other forms, such as heat or sound, due to factors like deformation and friction during the collision.
Please note that in solving the above questions, we need to use appropriate units, such as kilograms for mass and meters per second for velocity.