Question

(a) What is the kinetic energy in joules of an $850-\mathrm{lb}$ motorcycle moving at $66\mathrm{mph}?$ (b) By what factor will the kinetic energy change if the speed of the motorcycle is decreased to $33\mathrm{mph}?$ (c) Where does the kinetic energy of the motorcycle go when the rider brakes to a stop?

Short answer

The kinetic energy of an 850 lb motorcycle moving at 66 mph is approximately 166,405 Joules. When the speed is decreased to 33 mph, the kinetic energy decreases by a factor of 0.25. When the motorcycle comes to a stop, the kinetic energy is converted into heat due to friction between the brake pads and the wheel, and this heat is dissipated into the surrounding air.

Step-by-step solution

Convert mass to kilograms

To work with the SI (International System of Units), we first need to convert the mass of the motorcycle from pounds to kilograms. The conversion factor is 1 kg ≈ 2.205 lb. So, for an 850 lb motorcycle: $m=\frac{850\text{ lb}}{2.205\frac{\text{lb}}{\text{kg}}}=385.49\text{ kg}$

Convert speed to meters per second

Next, we convert the speed from miles per hour to meters per second. The conversion factors are 1 mile ≈ 1609.34 meters and 1 hour ≈ 3600 seconds. So, for a 66 mph speed: $v=\frac{66\text{mph}\cdot 1609.34\frac{\text{m}}{\text{mile}}}{3600\frac{\text{s}}{\text{hr}}}=29.48\frac{\text{m}}{\text{s}}$

Calculate the kinetic energy

Now, we can use the formula for kinetic energy, $KE=\frac{1}{2}m{v}^2$, to find the kinetic energy of the motorcycle. $KE=\frac{1}{2}(385.49\text{kg})(29.48\frac{\text{m}}{\text{s}}{)}^2=166404.73\text{J}$ The kinetic energy of the motorcycle moving at 66 mph is approximately 166,405 Joules.

Determine the change in kinetic energy with decreased speed

If the speed of the motorcycle is decreased to 33 mph, we first need to convert this to meters per second using the same conversion factors used in Step 2: $v^{\prime }=\frac{33\text{mph}\cdot 1609.34\frac{\text{m}}{\text{mile}}}{3600\frac{\text{s}}{\text{hr}}}=14.74\frac{\text{m}}{\text{s}}$ Now, recalculate the kinetic energy with the new speed: $K{E}^{\prime }=\frac{1}{2}(385.49\text{kg})(14.74\frac{\text{m}}{\text{s}}{)}^2=41601.18\text{J}$ To find the factor by which the kinetic energy changes, we can simply divide the new kinetic energy by the original kinetic energy: $\text{Factor}=\frac{K{E}^{\prime }}{KE}=\frac{41601.18\text{J}}{166404.73\text{J}}\approx 0.25$ The kinetic energy of the motorcycle will decrease by a factor of 0.25 when the speed is decreased to 33 mph.

Explain where the kinetic energy goes when the motorcycle brakes to a stop

When the rider applies the brakes to bring the motorcycle to a stop, the kinetic energy is converted into heat due to the friction between the brake pads and the wheel. This heat is then dissipated into the surrounding air, resulting in the loss of kinetic energy for the motorcycle.

Key concepts

Energy Conversion

When discussing energy, it's essential to understand that it can take various forms and can transform from one to another—a concept known as energy conversion. In the context of a moving motorcycle, as presented in the exercise, the vehicle possesses kinetic energy, which is the energy of motion. This kinetic energy is calculated based on the mass and velocity of the motorcycle using the formula:

$$KE=\frac{1}{2}m{v}^2$$ Where KE represents kinetic energy, m is the mass, and v is the velocity. When the motorcycle slows down by applying brakes, kinetic energy does not disappear; rather, it is converted into thermal energy due to friction between the brake pads and the wheels. This conversion aligns with the law of conservation of energy, which states that energy cannot be created or destroyed, only transformed from one form to another. The generated heat is eventually released into the environment, signifying a shift from mechanical to thermal energy.

SI Units Conversion

Another key area to consider is SI units conversion, especially when tackling physics problems. Working with standardized units allows for consistent measurements and calculations. The International System of Units (SI) specifies the kilogram (kg) for mass and the meter per second (m/s) for velocity. In the given exercise, to compute kinetic energy using SI units, we first convert the motorcycle's mass from pounds (lb) to kilograms (kg) and its speed from miles per hour (mph) to meters per second (m/s).

When converting units, we use conversion factors:
- 1 lb ≈ 0.453592 kg
- 1 mph ≈ 0.44704 m/s
These conversion factors are essential for translating the given quantities into the appropriate SI units to apply the kinetic energy formula correctly. Without proper unit conversion, our results would be inaccurate, leading to misunderstandings in the problem-solving process.

Work-Energy Principle

The work-energy principle is a fundamental concept that relates the work done on an object to its energy change. According to this principle, the work done by the net force acting on an object results in a change in the object's kinetic energy. In our motorcycle example, as the bike accelerates or decelerates, the work done by external forces, like friction and air resistance, converts kinetic energy into other energy forms or vice versa.

This principle provides valuable insights into the question of where the motorcycle's kinetic energy goes when it stops. The work done by friction (a resistive force) when the brakes are applied is negative, and this work equals the loss in kinetic energy. Formally, we express this as:
$$W=\mathrm{\Delta }KE$$ Where W is the work done and $\mathrm{\Delta }KE$ is the change in kinetic energy. This relationship ensures that the entire process adheres to the conservation laws of physics and equips us with a quantitative method to assess dynamic systems.