If the velocity of an object quadruples while falling, by what factor would the kinetic energy and momentum increase, respectively?
a) 4x and 4x
b) 4x and 2x
c) 16x and 4x
d) 16x and 16x
The correct answer here is C -- the kinetic energy increases by a factor of 16 while the momentum only increases by a factor of 4.
The formula for kinetic energy is
k = 1/2mv^2 or k = .5*m*v*v
where m = mass and v = velocity.
Since the mass of the object remains constant as it falls, only the velocity changes on the right side of this equation. When the velocity goes up by 4 times, the kinetic energy goes up by 16 times because the velocity gets squared.
The formula for momentum is
momentum = mass*velocity
Here, velocity is not squared so multiplying the velocity by 4 only increases the momentum by a factor of 4.
Momentum is the total motion possessed by a body. Mathematically momentum can be represented by the formula:
Momentum = Mass*Velocity
This means that the momentum is directly proportional to the speed or velocity. Thus if the speed of an object is increased n times, its momentum will also be increased n times.
Kinetic energy of an objects refers to the energy that it possesses because of its motion, or velocity. This kinetic energy can be calculated by the formula:
Kinetic energy = (mass*Velocity^2)/2
This means that the kinetic energy is directly proportional to the square of speed or velocity. Thus if the speed of an object is increased n times, its kinetic energy will be increased n^2 times.
In the question given above the velocity of the object is increase 4 times. Therefore its momentum will also increase 4 times, and its kinetic energy will increase 4^2 times - that is 16 times.
Therefore option c) of the multiple choice is correct.
Let the initial velocity of the object be v. At this velocity its kinetic energy, k1 is given by: k1 =(1/2) mv^2, where m is the mass of the object .
It's momentumP1 is given by ;P1 = mv.
Velocity when quadruple is 4v.
The kinetic energy k2 of the object at 4v velocity: k2 =(1/2)m(4v)^2 = (16/2)mv^2
The momentum P2 of the body at 4v velocity : P2 = m*(4v) =4mv
Therefore, by the above results we conclude that (c) is the correct choice.