Question

An advertisement claims that a certain \(1200-\mathrm{kg}\) car can accelerate from rest to a speed of \(25 \mathrm{~m} / \mathrm{s}\) in \(8.0 \mathrm{~s}\). What average power must the motor supply in order to cause this acceleration? Ignore losses due to friction.

Short answer

Answer: The average power the motor must supply in order to cause this acceleration is 46,875 W.

Step-by-step solution

Calculate the acceleration

To find the acceleration, we will use the formula: Acceleration (\(a\)) = (Final velocity (\(v_f\)) - Initial velocity (\(v_i\))) / Time (\(t\)) Given: Final velocity, \(v_f = 25 \mathrm{~m} / \mathrm{s}\) Initial velocity, \(v_i = 0 \mathrm{~m} / \mathrm{s}\) (since the car starts from rest) Time, \(t = 8.0 \mathrm{~s}\) Now, let's calculate the acceleration: \(a = (25 \mathrm{~m/s} - 0 \mathrm{~m/s}) / 8.0 \mathrm{~s}\) \(a = 25 / 8\) \(a = 3.125 \mathrm{~m} / \mathrm{s^2}\)

Calculate the force

Now that we have the acceleration, we can calculate the force using Newton's second law of motion: Force (\(F\)) = Mass (\(m\)) × Acceleration (\(a\)) Given: Mass, \(m = 1200 \mathrm{~kg}\) Acceleration, \(a = 3.125 \mathrm{~m} / \mathrm{s^2}\) Now, let's calculate the force: \(F = 1200 \mathrm{~kg} \times 3.125 \mathrm{~m} / \mathrm{s^2}\) \(F = 3750 \mathrm{~N}\)

Calculate the average power

Finally, we can calculate the average power supplied by the motor using the formula: Average power (\(P_{avg}\)) = Force (\(F\)) × Distance (\(d\)) / Time (\(t\)) We can find the distance using the formula for the average velocity: Average velocity (\(v_{avg}\)) = (Initial velocity (\(v_i\)) + Final velocity (\(v_f\))) / 2 \(v_{avg} = (0 \mathrm{~m/s} + 25 \mathrm{~m/s}) / 2\) \(v_{avg} = 12.5 \mathrm{~m/s}\) Now, we can calculate the distance using the average velocity and time: Distance (\(d\)) = Average velocity (\(v_{avg}\)) × Time (\(t\)) \(d = 12.5 \mathrm{~m/s} \times 8.0 \mathrm{~s}\) \(d = 100 \mathrm{~m}\) Now, let's calculate the average power: \(P_{avg} = 3750 \mathrm{~N} \times 100 \mathrm{~m} / 8.0 \mathrm{~s}\) \(P_{avg} = 375000 / 8\) \(P_{avg} = 46875 \mathrm{~W}\) The average power the motor must supply in order to cause this acceleration is \(46875 \mathrm{~W}\).

Key concepts

Newton's Second Law

Newton's Second Law is a fundamental principle in physics that describes the relationship between the motion of an object and the forces acting upon it. According to this law, the force acting on an object is equal to the mass of that object multiplied by its acceleration. This is expressed mathematically as: \[ F = ma \] where \( F \) is the force in newtons, \( m \) is the mass in kilograms, and \( a \) is the acceleration in meters per second squared.In the context of the car exercise, the car's mass is given as \( 1200 \mathrm{~kg} \), and the acceleration was calculated as \( 3.125 \mathrm{~m/s^2} \). By plugging these values into the formula, we can determine the force required to produce that acceleration. The resulting force is crucial in calculating the power needed for the car to reach its given speed.

Acceleration

Acceleration is the rate at which an object changes its velocity. It is a vector quantity, meaning it has both magnitude and direction. Acceleration occurs whenever there's a change in speed or direction. To calculate acceleration in this scenario, you use the formula:\[ a = \frac{v_f - v_i}{t} \] where \( v_f \) is the final velocity, \( v_i \) is the initial velocity, and \( t \) is the time taken for the change. For the car in the exercise, the initial velocity is zero because it starts from rest. Its final velocity is \( 25 \mathrm{~m/s} \), and the time taken is \( 8.0 \mathrm{~s} \). This leads to an acceleration of \( 3.125 \mathrm{~m/s^2} \). Understanding this concept helps you see how quickly an object can pick up speed under a certain force.

Average Velocity

Average velocity helps us understand how fast, on average, an object is moving over a given period. It is important when calculating other values, such as distance or power needed for motion. The average velocity can be calculated when you know the initial and final velocities:\[ v_{avg} = \frac{v_i + v_f}{2} \] For the car starting from rest in the exercise, the initial velocity (\( v_i \)) is \( 0 \mathrm{~m/s} \), and the final velocity (\( v_f \)) is \( 25 \mathrm{~m/s} \). Using the above formula, the average velocity turns out to be \( 12.5 \mathrm{~m/s} \).This average velocity is then used to calculate the distance traveled over \( 8.0 \mathrm{~s} \), which becomes handy when finding out how much work is performed during the acceleration phase. Understanding average velocity also allows us to appreciate how the car gradually attains its top speed.