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, \(v_{\text {ave }}=80.0 \mathrm{~km} / \mathrm{h}(22.2 \mathrm{~m} / \mathrm{s})\). Assume that energy is lost at a constant rate during the deceleration.

Short answer

Answer: The power needed to keep the car moving at a constant speed of 80.0 km/h is approximately 20,714 W (watts).

Step-by-step solution

Calculate the deceleration rate

To calculate the deceleration rate, we can use the following formula that relates initial velocity, final velocity, acceleration, and time: a = (v - v0) / t where a is the deceleration or acceleration, v0 and v are initial and final velocities, and t is the given time. In our case, we have: v0 = 25.0 m/s v = 19.4 m/s t = 6.00 s Plugging these values into the formula, we get: a = (19.4 - 25.0) / 6.00 a = -5.6/6.00 a ≈ -0.933 m/s² The negative sign indicates the car is decelerating.

Calculate the force acting on the car

Now that we have the deceleration rate, we can calculate the force acting on the car using Newton's second law: F = m * a where F is the force acting on the car, m is the mass of the car, and a is the deceleration rate. In our case, we have: m = 1000 kg a = -0.933 m/s² Plugging these values into the formula, we get: F = 1000 * (-0.933) F ≈ -933 N The negative sign indicates that the force is acting opposite to the direction of motion, causing the car to slow down.

Calculate the power needed to maintain a constant speed

To calculate the power needed to maintain a constant speed, we can use the following formula that relates power, force, and velocity: P = F * v_ave where P is the power, F is the force acting on the car, and v_ave is the average velocity at which the car is supposed to maintain a constant speed. In our case, we have: F = -933 N (we will consider the magnitude of the force, so we drop the negative sign) v_ave = 22.2 m/s (80.0 km/h) Plugging these values into the formula, we get: P = 933 * 22.2 P ≈ 20714 Watts Therefore, the power needed to keep the car moving at a constant speed of 80.0 km/h (22.2 m/s) is approximately 20,714 W (watts).

Key concepts

Deceleration Rate Calculation

Deceleration is simply acceleration in the opposite direction of motion. In physics, this phenomenon, along with acceleration, is described by the change in velocity of an object over a period of time. For a driver slowing down from a higher to a lower speed, it's vital to calculate the rate of deceleration to understand the forces at play.

To calculate deceleration, which is simply negative acceleration, we use the formula: \( a = \frac{v - v_0}{t} \). Here, \( a \) represents the deceleration, \( v_0 \) is the initial velocity, \( v \) is the final velocity, and \( t \) is time. The negative value points out that the car is slowing down rather than speeding up.

Understanding deceleration is crucial, not just for problem-solving in physics but also for real-world applications like designing braking systems in vehicles or calculating stopping distances for safety.

Newton's Second Law

One of the cornerstones of classical mechanics is encapsulated in Newton's second law of motion. It states that the force acting on an object is equal to the mass of that object multiplied by its acceleration (\( F = m \times a \)). This law forms the basis for a wide array of calculations involving motion.

In the case of a decelerating car, we employ this law to determine the force required to bring the vehicle to a slower speed. Using the previously calculated deceleration rate and the known mass of the car, we apply Newton's second law to find the exact magnitude of the force exerted against the motion of the car.

Newtons's second law is integral not just for solving physics problems but also for practical engineering. From designing vehicles to planning spacecraft trajectories, Newton's insight continues to be a fundamental tool in understanding and applying the principles of forces and motion.

Power Calculation

Power in physics is defined as the rate at which work is done or energy is transferred. When calculating the power required to maintain a constant speed against a decelerating force, we use the formula: \( P = F \times v_{\text{ave}} \), where \( P \) is power, \( F \) is force, and \( v_{\text{ave}} \) is average velocity.

In the scenario where a car is attempting to maintain a steady pace, knowing the power gives insight into the engine's performance requirements. By considering the magnitude of the force acting on the car and the desired constant velocity, we can establish the continual output needed from the car's engine.

The concept of power is not only vital in problem-solving within physics but extends its importance to understanding and designing anything from simple machines to complex energy systems. This blend of force and movement is often seen in everyday life, particularly in the automotive industry, where it dictates the efficiency and capability of the engine.