A motorist driving at 25 meters/second decelerates to 15 meters/second. The combined mass of the vehicle and the driver is [tex]$1.5 \times 10^3$[/tex] kilograms. What is the amount of mechanical work done by the vehicle?
A. [tex]$7.5 \times 10^4$[/tex] joules
B. [tex]$1.5 \times 10^5$[/tex] joules
C. [tex]$3.0 \times 10^5$[/tex] joules
D. [tex]$6.2 \times 10^5$[/tex] joules
Answer (1)
Calculate the initial kinetic energy: K E i = 2 1 m v i 2 = 4.6875 × 1 0 5 J.
Calculate the final kinetic energy: K E f = 2 1 m v f 2 = 1.6875 × 1 0 5 J.
Calculate the change in kinetic energy: Δ K E = K E f − K E i = − 3.0 × 1 0 5 J.
The mechanical work done is equal to the change in kinetic energy: W = − 3.0 × 1 0 5 J. The amount of work done is 3.0 × 1 0 5 joules.
Explanation
Problem Analysis We are given the initial velocity v i = 25 m/s, the final velocity v f = 15 m/s, and the combined mass m = 1.5 × 1 0 3 kg. We need to find the mechanical work done by the vehicle. The mechanical work done is equal to the change in kinetic energy.
Calculating Initial Kinetic Energy First, we calculate the initial kinetic energy K E i using the formula K E = 2 1 m v 2 . Thus, K E i = 2 1 m v i 2 = 2 1 ( 1.5 × 1 0 3 kg ) ( 25 m/s ) 2 = 2 1 ( 1.5 × 1 0 3 ) ( 625 ) = 0.75 × 1 0 3 × 625 = 468.75 × 1 0 3 J = 4.6875 × 1 0 5 J
Calculating Final Kinetic Energy Next, we calculate the final kinetic energy K E f using the same formula: K E f = 2 1 m v f 2 = 2 1 ( 1.5 × 1 0 3 kg ) ( 15 m/s ) 2 = 2 1 ( 1.5 × 1 0 3 ) ( 225 ) = 0.75 × 1 0 3 × 225 = 168.75 × 1 0 3 J = 1.6875 × 1 0 5 J
Calculating Change in Kinetic Energy Now, we find the change in kinetic energy Δ K E = K E f − K E i :
Δ K E = K E f − K E i = 1.6875 × 1 0 5 J − 4.6875 × 1 0 5 J = − 3.0 × 1 0 5 J
Determining Mechanical Work Done The mechanical work done W is equal to the change in kinetic energy Δ K E . Therefore, W = Δ K E = − 3.0 × 1 0 5 J The work done by the vehicle is − 3.0 × 1 0 5 joules. The negative sign indicates that the work is done by the vehicle to decelerate, which means the vehicle loses kinetic energy.
Final Answer The amount of mechanical work done by the vehicle is − 3.0 × 1 0 5 joules. Therefore, the correct answer is C. 3.0 × 1 0 5 joules. Note that the question asks for the amount of work done, so we take the absolute value.
Examples
Understanding the concept of mechanical work is crucial in analyzing vehicle motion. For instance, when designing a braking system, engineers need to calculate the amount of work the brakes must do to bring a vehicle to a stop from a certain speed. This calculation involves the change in kinetic energy, similar to the problem we solved. By accurately determining the work required, engineers can design effective and safe braking systems that ensure vehicles can decelerate reliably under various conditions. This principle extends to other aspects of vehicle design, such as optimizing fuel efficiency and understanding the impact of different driving conditions on vehicle performance.
