An 1100-kg car is traveling at 46 km/h on a level road. The brakes are applied long enough to remove 51 kJ of kinetic energy. (a) What is the final speed of the car? (b) How much more kinetic energy must be removed by the brakes to stop the car?

To tackle the problem of the car's kinetic energy and its final speed after the brakes are applied, we need to start with some fundamental physics concepts. Kinetic energy (KE) is determined by the formula KE = 0.5 * m * v², where m is the mass of the object and v is its velocity. Let's break the problem down into manageable parts.

Part (a): Finding the Final Speed of the Car

The first step is to calculate the initial kinetic energy of the car before the brakes are applied. Given that the car has a mass (m) of 1100 kg and is traveling at a speed (v) of 46 km/h, we first need to convert the speed into meters per second (m/s) for consistency with SI units. The conversion from km/h to m/s is as follows:

• 46 km/h = 46 * (1000 m / 1 km) * (1 h / 3600 s) = 12.78 m/s

Now, we can calculate the initial kinetic energy:

• KE_initial = 0.5 * m * v² = 0.5 * 1100 kg * (12.78 m/s)²
• KE_initial = 0.5 * 1100 * 163.3 ≈ 89,815 J (joules)

Next, we know that the brakes remove 51 kJ of kinetic energy, which is equivalent to:

• 51 kJ = 51,000 J

To find the final kinetic energy after the brakes are applied, we subtract the energy removed:

• KE_final = KE_initial - Energy_removed
• KE_final = 89,815 J - 51,000 J = 38,815 J

Now that we have the final kinetic energy, we can find the final speed (v_final) of the car using the kinetic energy formula:

• KE_final = 0.5 * m * v_final²
• 38,815 J = 0.5 * 1100 kg * v_final²

Solving for v_final, we rearrange the equation:

• v_final² = (2 * KE_final) / m
• v_final² = (2 * 38,815 J) / 1100 kg
• v_final² = 70.56
• v_final ≈ 8.4 m/s

Thus, the final speed of the car after applying the brakes is approximately 8.4 m/s.

Part (b): Finding the Additional Kinetic Energy Needed to Stop the Car

The next part of the question asks how much more kinetic energy must be removed to bring the car to a complete stop. When the car is at rest, its kinetic energy is 0 J. Therefore, the additional kinetic energy that needs to be removed is simply the final kinetic energy we calculated:

• Energy_to_remove = KE_final = 38,815 J

To express this in kilojoules, we convert joules to kilojoules:

• Energy_to_remove = 38,815 J ÷ 1000 = 38.82 kJ

In conclusion, the car needs to remove an additional 38.82 kJ of kinetic energy to come to a complete stop.