Which statement best compares momentum and kinetic energy?
A. If you double the velocity of an object, its kinetic energy doubles. But for the same increase in velocity, the momentum increases four times.
B. If you double the velocity of an object, its momentum doubles. But for the same increase in velocity, the kinetic energy increases four times.
C. If you double the velocity of an object, its momentum and kinetic energy doubles.
D. If you double the velocity of an object, its momentum and kinetic energy increases four times.
Answer (1)
Momentum doubles when velocity doubles: p = m v , so if v → 2 v , then p → 2 m v .
Kinetic energy increases four times when velocity doubles: K E = 2 1 m v 2 , so if v → 2 v , then K E → 2 1 m ( 2 v ) 2 = 2 m v 2 , which is four times the original KE.
Therefore, the correct statement is that if you double the velocity of an object, its momentum doubles, and its kinetic energy increases four times.
The answer is B.
Explanation
Problem Analysis Let's analyze how momentum and kinetic energy change when the velocity of an object is doubled. We'll use the definitions of momentum and kinetic energy to determine the correct statement.
Definitions of Momentum and Kinetic Energy Momentum (p) is defined as the product of mass (m) and velocity (v): p = m v Kinetic energy (KE) is defined as one-half the product of mass (m) and the square of velocity (v): K E = 2 1 m v 2
Initial Momentum and Kinetic Energy Let's consider an object with mass m moving at an initial velocity v 1 = v . Its initial momentum p 1 and kinetic energy K E 1 are: p 1 = m v K E 1 = 2 1 m v 2
Momentum and Kinetic Energy with Doubled Velocity Now, let's double the velocity to v 2 = 2 v . The new momentum p 2 and kinetic energy K E 2 are: p 2 = m ( 2 v ) = 2 m v K E 2 = 2 1 m ( 2 v ) 2 = 2 1 m ( 4 v 2 ) = 2 m v 2
Comparing the Changes Let's compare the new momentum and kinetic energy to the initial values. The ratio of the new momentum to the initial momentum is: p 1 p 2 = m v 2 m v = 2 So, when the velocity doubles, the momentum also doubles. The ratio of the new kinetic energy to the initial kinetic energy is: K E 1 K E 2 = 2 1 m v 2 2 m v 2 = 4 So, when the velocity doubles, the kinetic energy increases by a factor of 4.
Conclusion Based on our calculations, when the velocity of an object doubles, its momentum doubles, and its kinetic energy increases four times. This corresponds to option B.
Examples
Understanding the relationship between velocity, momentum, and kinetic energy is crucial in many real-world scenarios. For example, in car safety, doubling the speed of a car doesn't just double the impact force (momentum) in a collision; it quadruples the energy that needs to be absorbed by the car's safety features (kinetic energy). This is why even small increases in speed can significantly increase the risk of serious injury in a car accident. Similarly, in sports, understanding these relationships helps athletes optimize their performance and safety. For instance, a baseball player knows that increasing the bat's speed will have a much greater impact on the ball's kinetic energy (and thus how far it travels) than on its momentum.
