Consider two cars, a very large one like a Hummer (mass 5m) and a small one like a Mini Cooper (mass m), both starting at the same point race toward the edge of a cliff. Assume that each car is accelerated by a constant force (same size force on each car) to the edge of the cliff.
A. which car has the greater acceleration?
B. Which car takes longer to reach the edge?
C. Which car has the larger impulse imparted to it by the applied force?
D. Which car has the greater momentum at the edge ?
E. WHich car has the greater work done on it?
F. Which car has the greater kinetic energy at the edge of the cliff?
A. From F=ma i'd say it would be the hummer as its got greater mass, and as Force is constant
A) With F=ma you get m=F/a. If F is a constant then; when mass increases the acceleration decreases and vice-versa (it's reversely proportional). So the Mini Cooper with lower mass has a greater acceleration.
B) With the above conclusion it is sensible that the Hummer takes a longer time.
C) Well impulse is measured by F*t. As F is a constant the Hummer which consumes more time(low acceleration) has a larger impulse.
D) We know that the impulse equals to the change of momentum. So it's the Hummer that has the greater momentum.
E) Work is the Force into the displacement of the center of mass. Here the Force is a constant and the displacement is the same for both cars. So it's reasonable enough to say that the work done on each car is equal.
F) Considering the work done is same for both cars, you easily get the answer that the Kinetic energies of the two cars are equal.
To answer these questions, we need to use Newton's second law of motion, which states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The equation for this relationship is F = ma, where F is the force, m is the mass, and a is the acceleration.
A. Which car has the greater acceleration?
According to Newton's second law, the acceleration is inversely proportional to the mass. Therefore, the smaller car (Mini Cooper) with mass m will have a greater acceleration compared to the larger car (Hummer) with mass 5m.
B. Which car takes longer to reach the edge?
To determine the time taken by each car to reach the edge, we need additional information - the magnitude of the constant force applied to each car. Assuming that the forces are the same, both cars will experience the same force. Therefore, since the smaller car has a greater acceleration, it will take less time to reach the edge compared to the larger car.
C. Which car has the larger impulse imparted to it by the applied force?
Impulse is calculated as the product of force and time, J = Ft. Since both cars experience the same force, the one that takes longer (which is the larger car in this case) will have a larger impulse imparted to it.
D. Which car has the greater momentum at the edge?
Momentum is given by the product of mass and velocity, p = mv. Assuming that both cars start from rest and reach the same final velocity at the edge of the cliff, the car with the larger mass (Hummer) will have a greater momentum compared to the smaller car (Mini Cooper).
E. Which car has the greater work done on it?
Work done is equal to the product of force and displacement, W = Fd. Since both cars are subjected to the same constant force, the work done on each car will depend on the distance traveled (displacement). Without knowing the distance they travel, we cannot determine which car has greater work done on it.
F. Which car has the greater kinetic energy at the edge of the cliff?
Kinetic energy is given by the equation K.E. = 1/2 mv^2, where m is the mass and v is the velocity. Assuming both cars start from rest and reach the same final velocity at the edge, the car with the larger mass (Hummer) will have a greater kinetic energy than the smaller car (Mini Cooper) due to its higher mass.