Question

A driver notices that her 1000 , \(\mathrm{kg}\) car slows from \(v_{0}=\) \(90.0 \mathrm{~km} / \mathrm{h}(25.0 \mathrm{~m} / \mathrm{s})\) to \(v=70.0 \mathrm{~km} / \mathrm{h}(19.4 \mathrm{~m} / \mathrm{s})\) in \(t=6.00 \mathrm{~s}\) moving on level ground in neutral gear. Calculate the power needed to keep the car moving at a constant speed, \(v_{\text {ave }}=\) \(80.0 \mathrm{~km} / \mathrm{h}(22.2 \mathrm{~m} / \mathrm{s})\)

Short answer

Answer: The power required to keep the car at a constant speed is 20.7 kW.

Step-by-step solution

1. Calculate the acceleration of the car

To calculate the force acting on the car, we first need to determine the acceleration. We can find this by using the formula: acceleration (\(a\)) \(= \frac{v - v_0}{t}\). From the given information, \(v_0 = 25.0 \ \text{m/s}\), \(v = 19.4 \ \text{m/s}\), and \(t = 6.00 \ \text{s}\). Plug in the known values: \(a = \frac{19.4 - 25.0}{6.00}\).

2. Calculate the car's deceleration

After calculating the acceleration, we find that \(a = -0.933 \ \text{m/s}^2\). Since the acceleration is negative, this implies that the car is decelerating.

3. Calculate the force acting on the car

Now that we have the deceleration, we can calculate the force (\(F_d\)) acting on the car using Newton's second law, \(F_d = m \times a\), where \(m\) is the mass of the car, which is given as \(1000 \ \text{kg}\). Plug in the known values: \(F_d = 1000 \times (-0.933)\).

4. Determine the decelerating force

After calculating the force, we get \(F_d = -933 \ \text{N}\). The negative sign here indicates that this is a decelerating force, which is what we expect.

5. Calculate the power required to maintain constant speed

To maintain a constant speed, the car needs to exert an equal and opposite force against the decelerating force. So, the required force (\(F_r\)) is \(F_r = -F_d = 933 \ \text{N}\). Now we can use the formula for power (\(P\)): \(P = F_r \times v_{\text{ave}}\), where \(v_{\text{ave}}\) is the average velocity, given as \(22.2 \ \text{m/s}\). Plug in the known values: \(P = 933 \times 22.2\).

6. Calculate the required power

After performing the calculations, we find that the required power to maintain a constant speed is \(P = 20,700 \ \text{W}\) or \(20.7 \ \text{kW}\).

Key concepts

Acceleration and Deceleration

When we talk about acceleration and deceleration, we are referring to changes in the speed of an object. Acceleration is the rate at which an object speeds up, while deceleration is the rate at which an object slows down. Even though we often think of acceleration as speeding up, in physics, it refers to any change in velocity, which means deceleration is technically negative acceleration.

To calculate the acceleration or deceleration, we use the formula:
\[a = \frac{v - v_0}{t}\]
where \(v\) is the final velocity, \(v_0\) is the initial velocity, and \(t\) is the time over which the change occurs. The sign of the acceleration will tell us if the object is speeding up (positive acceleration) or slowing down (negative acceleration).

In our exercise, the car decelerates because its final velocity is less than its initial velocity, leading to a negative acceleration. Understanding the concept of deceleration is crucial when analyzing motion and forces acting on objects.

Newton's Second Law

At the heart of motion and force interactions lies Newton's second law of motion. Newton's second law states that the force acting on an object is equal to the mass of the object multiplied by its acceleration. Mathematically, it can be expressed as:
\[F = m \times a\]
where \(F\) is the force, \(m\) is the mass, and \(a\) is the acceleration. This law is fundamental when we want to determine the force required to cause a certain acceleration, or conversely, to find out the acceleration that a certain force will produce.

In the context of our textbook exercise, we apply Newton's second law to calculate the force of deceleration acting on the car. Once the deceleration is known, we can find the force needed to move the car at a constant speed by simply reversing the direction of the deceleration force, highlighting how forces can be manipulated to achieve a desired motion.

Power Calculation

Power in physics is defined as the rate at which work is done or energy is transferred. When dealing with moving objects, power calculation tells us how fast a force is doing work. The equation for power when force and velocity are given is:
\[P = F \times v\]
where \(P\) is power, \(F\) is the force, and \(v\) is the velocity or speed at which the force is applied. The resultant unit for power is Watts (W), which represents joules per second.

In the exercise we're dissecting, calculating the power required to maintain constant speed involves using the force that would counteract the deceleration (the equal and opposite force) and the average velocity of the car. This provides a practical application of power calculations, helping us understand how much energy is needed to keep a vehicle moving at a steady pace in real-world scenarios.