Select the correct answer.
A car has a kinetic energy of [tex]$1.9 \times 10^3$[/tex] joules. If the velocity of the car is decreased by half, what is its kinetic energy?
A. [tex]$2.2 \times 10^2$[/tex] joules
B. [tex]$4.7 \times 10^2$[/tex] joules
C. [tex]$5.2 \times 10^2$[/tex] joules
D. [tex]$5.8 \times 10^3$[/tex] joules
Answer (2)
The initial kinetic energy is K E 1 = 1.9 × 1 0 3 joules.
The velocity is halved, so v 2 = 2 1 v 1 .
The new kinetic energy is K E 2 = 4 1 K E 1 .
Calculate K E 2 = 4 1 ( 1.9 × 1 0 3 ) = 4.75 × 1 0 2 joules, so the answer is 4.7 × 1 0 2 joules.
Explanation
Problem Analysis We are given that a car has a kinetic energy of 1.9 × 1 0 3 joules. The velocity of the car is decreased by half, and we want to find the new kinetic energy.
Kinetic Energy Formula The formula for kinetic energy is given by K E = 2 1 m v 2 , where K E is the kinetic energy, m is the mass, and v is the velocity.
Initial Kinetic Energy Let K E 1 be the initial kinetic energy and v 1 be the initial velocity. Then K E 1 = 2 1 m v 1 2 = 1.9 × 1 0 3 J.
New Kinetic Energy The velocity is decreased by half, so the new velocity is v 2 = 2 1 v 1 . The new kinetic energy K E 2 is given by K E 2 = 2 1 m ( 2 1 v 1 ) 2 .
Relating New and Initial Kinetic Energies We can rewrite K E 2 as K E 2 = 2 1 m ( 4 1 v 1 2 ) = 4 1 ( 2 1 m v 1 2 ) = 4 1 K E 1 .
Calculating New Kinetic Energy Substituting the value of K E 1 , we get K E 2 = 4 1 ( 1.9 × 1 0 3 ) = 4.75 × 1 0 2 joules.
Final Answer The new kinetic energy is 4.75 × 1 0 2 joules, which is approximately 4.7 × 1 0 2 joules. Therefore, the correct answer is B.
Examples
Understanding kinetic energy changes is crucial in vehicle safety. For instance, if a car's velocity is halved before a collision, its kinetic energy reduces to one-fourth, significantly lessening the impact force. This principle is applied in designing crumple zones in cars, which absorb energy during a crash, reducing the force experienced by the occupants. Similarly, in sports, understanding how velocity affects kinetic energy helps athletes optimize their performance and minimize injury risks.
The new kinetic energy of the car, after its velocity is decreased by half, is calculated to be approximately 4.7 × 1 0 2 joules. This is found by recognizing that halving the velocity reduces the kinetic energy to a quarter of its original value. Therefore, the correct answer is B.
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