Question

A car travels up a hill at the constant speed of \(40 \mathrm{~km} / \mathrm{h}\) and returns down the hill at the speed of \(60 \mathrm{~km} / \mathrm{h} .\) Calculate the average speed for the round trip.

Short answer

The average speed for the round trip is \(48 \mathrm{km/h}\).

Step-by-step solution

Calculate Time for Each Trip

The first step is to calculate the time it took for each trip. The distance up the hill is identical to the distance down. Define that distance as \(d\). Hence, the time to get to the top, \(t_{up}\), is \(d/40\) hours, as speed is distance/time. Similarly, the time to go down, \(t_{down}\), is \(d/60\) hours.

Calculate the Total Time

Next, calculate the total time for the round trip by summing the time up and the time down. That is, \(t_{total} = t_{up} + t_{down} = d/40 + d/60\). This simplifies to \(t_{total} = 5d/120 = d/24\) hours.

Calculate The Total Distance

The total distance covered for the round trip is two times the distance to the top of the hill. Hence, \(d_{total} = 2d\).

Calculate the Average Speed

The average speed is the total distance over the total time. Substituting the values obtained from steps 2 and 3, \(v_{avg} = d_{total}/t_{total} = (2d)/(d/24) = 48 \mathrm{km/h}\).

Key concepts

Uniform Motion

Uniform motion refers to the concept in physics where an object moves at a constant speed along a straight path. In the context of our exercise, the car traveling up and down the hill at constant speeds of 40 km/h and 60 km/h, respectively, exhibits uniform motion. This is important because uniform motion allows us to define specific times for the car's journey up and down based on distance and speed. Here, we do not have to consider acceleration or deceleration, which simplifies calculations substantially.

For instance, consider a car moving at a steady 50 km/h for an hour. It will cover a distance of 50 km. If it maintained that speed for 2 hours, it would cover 100 km. The certainty of distance covered over time is a hallmark of uniform motion and is particularly helpful in solving problems related to average speed over a distance, as seen in the exercise.

Kinematics

Kinematics is the branch of classical mechanics that deals with the description of motion without considering the causes of this motion, such as forces or mass. It encompasses the concepts of position, velocity, acceleration, and time. The kinematic equations relate these variables in a way that allows us to predict how an object will move based on its current state.

In the textbook exercise, kinematics come into play when we calculate the time taken for the car to travel up and down the hill. These times are crucial for determining average speed. While the exercise does not include acceleration – the car travels at a uniform speed – the principles of kinematics still guide the relationship between the distance traveled, the constant speed, and the time taken, as shown in the step-by-step solution.

Speed and Velocity

The concepts of speed and velocity are fundamental to understanding motion.

Speed

Speed is a scalar quantity that indicates how fast an object is moving. It is calculated by dividing the distance traveled by the time taken to cover that distance. In the original exercise, we calculate the car's speed when going up and down the hill.

Velocity

Velocity, on the other hand, is a vector quantity, which means it has both magnitude and direction. It tells us the speed of the object as well as the direction of its movement.

The average speed for any trip is determined by dividing the total distance traveled by the total time taken, regardless of the direction of travel. This is what we calculate in the final step of the solution: by knowing the times for each leg of the trip and the uniform speeds at which the car travels, we can ascertain the average speed for the entire round trip journey.