Question

A \(3700-\mathrm{lb}\) automobile \((m=1600 \mathrm{~kg})\) starts from rest on a level road and gains a speed of \(45 \mathrm{mi} / \mathrm{h}(=72 \mathrm{~km} / \mathrm{h})\) in \(33 \mathrm{~s}\). (a) What is the kinetic energy of the auto at the end of the \(33 \mathrm{~s} ?(b)\) What is the average net power delivered to the car during the 33 -s interval? \((c)\) What is the instantaneous power at the end of the 33 -s interval assuming that the acceleration was constant?

Short answer

The kinetic energy of the automobile at the end of 33s is 323738.7264 Joules. The average power delivered to the car during the 33-s interval is 9810.266256 Watt and the instantaneous power at the end of 33s is also 9810.266256 Watt.

Step-by-step solution

Calculation of final velocity of the automobile

The final speed is given as 45 miles/hour, which needs to be converted to meters per second for ease of calculation. It can be done using the conversion relation \(1 \mathrm{mile / hour} = 0.44704 \mathrm{meters / second}\). The converted speed (v) comes out to be 20.1168 meters per second.

Calculation of kinetic energy

The kinetic energy (KE) of the automobile can be calculated using the relation \(KE = \frac{1}{2} m v^2\), where m is the mass of the automobile and v is its velocity. Substituting the known values, we get \(KE = \frac{1}{2} * 1600 \mathrm{kg} * (20.1168 \mathrm{m/s})^2 = 323738.7264 \mathrm{joules}\)

Calculation of average power

The average power (P_avg) can be calculated using the relation \(P_avg = \frac{KE}{t}\), where KE is the kinetic energy and t is the time. Substituting the known values, we get \(P_avg = \frac{323738.7264 \mathrm{joules}}{33 \mathrm{s}} = 9810.266256 \mathrm{watt}\)

Calculation of instantaneous power

Under the assumption of constant acceleration, the instantaneous power \(P_inst\) at the end of 33 sec is the same as the average power. Hence, \(P_inst = 9810.266256 watt\)

Key concepts

Physics Problem Solving

Understanding a physics problem involves identifying the known parameters and determining the proper formulas to apply. In our automobile example, the first step in problem-solving is converting units to a standard form, meters per second in this case, allowing for consistent calculations. The steps to solving physics problems often include:

• Identify the given information and the quantity you need to find.
• Create a plan by selecting the appropriate physical principles and formulas.
• Execute the plan by substituting known values into the formulas to find the quantity sought.
• Check the solution for consistency and accuracy.

For more clarity, visualize the situation through diagrams if possible and cross-check your answers with dimensional analysis to ensure they make sense both numerically and conceptually.

Kinetic Energy Formula

Kinetic energy, represented by the symbol KE, is the energy that an object possesses due to its motion. The formula for calculating the kinetic energy of an object is given by \[ KE = \frac{1}{2} m v^2 \]where 'm' stands for the mass of the object, and 'v' signifies its velocity. In our automobile scenario, after converting the car's speed into meters per second, we plug in the mass of the car and its speed into the equation to calculate its kinetic energy at the end of the acceleration period.
Understanding kinetic energy is crucial in various fields such as engineering, automobile safety, and physics. By knowing how to calculate kinetic energy, one can predict the impact force in a car crash or the required braking distance to stop a vehicle, emphasizing the practical application of this concept.

Average Power Calculation

Average power is calculated as the total work done by a force over an interval of time. Work, in turn, is synonymous with energy change. The formula to calculate average power \( P_{avg} \) is \[ P_{avg} = \frac{KE}{t} \]where KE represents the kinetic energy and 't' the time over which the work is done. Our example involves using the kinetic energy we've calculated over the specified 33-second interval to find the average power.
Understanding average power is vital for designing engines and motors as it describes the rate at which energy is converted from one form to another, such as fuel to kinetic energy in cars, helping us evaluate efficiency and performance.

Instantaneous Power Calculation

Instantaneous power is the power at any given moment in time and can be quite dynamic, especially if the speed of the object is changing rapidly. In cases of constant acceleration, instantaneous power at a specific time can equal the average power calculated over an interval. The formula for instantaneous power at a time 't' when acceleration is constant is given by \[ P_{inst} = F \times v \]since power is the dot product of force 'F' and velocity 'v'. In our scenario, we've simplified the calculation by equating average power with instantaneous power due to constant acceleration.
Knowing how to calculate instantaneous power is especially useful for understanding how engines and motors respond to varying conditions and for designing systems that function optimally across a broad range of operating conditions.