A person drives an automobile with a mass of 450 kilograms at a velocity of 26 meters per second. The driver accelerates to a velocity of 30 meters per second. The difference in the automobile’s kinetic energy between the two velocities is joules. You may use the calculator.
The correct answer and explanation is:
Let’s solve the problem step by step.
Given data:
- Mass, m=450 kgm = 450 \, \text{kg}
- Initial velocity, v1=26 m/sv_1 = 26 \, \text{m/s}
- Final velocity, v2=30 m/sv_2 = 30 \, \text{m/s}
Step 1: Write down the kinetic energy formula:
KE=12mv2KE = \frac{1}{2} m v^2
Step 2: Calculate initial kinetic energy:
KE1=12×450×(26)2=225×676=152,100 JoulesKE_1 = \frac{1}{2} \times 450 \times (26)^2 = 225 \times 676 = 152,100 \, \text{Joules}
Step 3: Calculate final kinetic energy:
KE2=12×450×(30)2=225×900=202,500 JoulesKE_2 = \frac{1}{2} \times 450 \times (30)^2 = 225 \times 900 = 202,500 \, \text{Joules}
Step 4: Find the difference in kinetic energy:
ΔKE=KE2−KE1=202,500−152,100=50,400 Joules\Delta KE = KE_2 – KE_1 = 202,500 – 152,100 = 50,400 \, \text{Joules}
Answer:
The difference in the automobile’s kinetic energy between the two velocities is 50,400 Joules.
Explanation:
Kinetic energy (KE) is the energy possessed by an object due to its motion. It depends on both the mass of the object and the square of its velocity, following the formula KE=12mv2KE = \frac{1}{2} m v^2. When the velocity of an object changes, its kinetic energy changes accordingly.
In this problem, the automobile’s velocity increases from 26 m/s to 30 m/s. Even though this change might seem small, because kinetic energy depends on the square of velocity, the energy difference can be substantial.
First, we calculate the kinetic energy at the initial velocity (26 m/s), then the kinetic energy at the final velocity (30 m/s). The difference between these two values represents the energy added to the car to increase its speed.
This difference in kinetic energy (50,400 Joules) essentially tells us how much work (energy) the engine must supply to accelerate the vehicle from 26 m/s to 30 m/s, neglecting other forces like friction or air resistance. Understanding this energy change is crucial in physics and engineering to analyze the performance and fuel efficiency of vehicles.