Question

You are in a bus traveling on a straight road ar \(20 \mathrm{~m} / \mathrm{s}\). As you pass a gas station, your clock and a clock in the station read precisely 0. You pass another gas station \(900 \mathrm{~m}\) farther down the road. ( In the frame of reference of the gas stations. all gas station clocks are synchronized.) (a) As you pass the second station. do you find its clock to be ahead of, or behind your own clock, and (b) by how much?

Short answer

a) The bus's clock is not ahead of or behind the gas station's clock. b) There is no difference in time; both the bus's clock and the gas station's clock read 45 seconds when the bus passes the second station.

Step-by-step solution

Understand the Scenario and Gather Information

Analyze the exercise and its specifics. The bus is moving at a speed of \(20 m/s\). We are considering two events: when the bus passes the first and the second gas station. The difference between these two events in the reference frame of the gas stations (which are stationary relative to each other) is \(900 m\). Time dilation and length contraction effects, important in the theory of relativity, will not be observable because this problem takes place at speeds much lower than the speed of light.

Calculation of Time for Bus and Gas Station

First, calculate the time it takes for the bus to travel from the first station to the second in the bus's frame of reference. To do this, use the formula for time, which is distance divided by speed. So, the time would be calculated as \(900 m / 20 m/s = 45 s\). This means that according to the bus's clock, 45 seconds have passed when it reaches the second station.

Determine the Time from Gas Station's Perspective

As the buses and stations are in relative motion, and we are considering much slower speeds than the speed of light, the time passed according to the gas stations is the same as calculated for the bus frame, that is, 45 seconds. In the frame of reference of the gas stations, all gas station clocks are synchronized. Therefore, when the bus reaches the second station, the clock at the gas station also reads 45 seconds.

Compare the Two Times

Compare the time on the bus's clock and the time on the second gas station's clock. They both read 45 seconds when the bus passes the second gas station. Therefore, as the bus passes the second station, it does not find its clock to be ahead of or behind the gas station clock. Both clocks read the same time.

Key concepts

Relative Motion

Relative motion refers to the movement of an object as observed from a particular frame of reference. In our scenario, the bus and the gas stations are in relative motion with each other. This means they are moving with respect to one another.

Let's look at it from the bus's point of view. The bus is moving at a speed of 20 m/s. As it travels down the road, it perceives the gas stations as moving towards it and then past it.

From the gas stations' point of view, the bus moves past them at the same speed. Understanding relative motion is essential because it defines how we interpret the movement of objects from different perspectives.

Frame of Reference

A frame of reference is a viewpoint (or coordinate system) used to measure and observe physical phenomena, like motion. In this problem, there are two frames of reference:

• The bus's frame of reference
• The gas stations' frame of reference

From the bus's frame, the gas stations are moving towards it. However, in the gas stations' frame of reference, they are stationary and the bus is the one moving. This dual frame of reference allows us to understand how measurements of time and distance depend on the observer's position and speed.

Time Synchronization

Time synchronization is crucial when comparing times between two different clocks located in separate frames of reference. In this situation, the clocks at both gas stations are synchronized. They read the same time when the bus passes each station.

In classical physics (unlike in relativity), time remains constant across different frames of reference moving at relatively low speeds, such as 20 m/s. This synchronization helps us compare the time readings accurately when the bus moves past each gas station.

Speed and Velocity

Speed and velocity are important concepts in understanding motion. Although they are often used interchangeably, they have distinct meanings.

• Speed is a scalar quantity, showing how fast an object is moving regardless of its direction. In this case, the bus's speed is 20 m/s.
• Velocity is a vector quantity, meaning it describes both the speed and direction of motion. If the bus moves in a straight line towards the second gas station, its velocity is also 20 m/s, directed along the road.

Understanding both speed and velocity helps define the motion and the calculations involved in determining distances traveled and time taken.

Kinematics

Kinematics is the branch of classical physics that studies motion without considering the forces that cause it. It focuses on aspects such as velocity, acceleration, displacement, and time.

In this exercise, we use kinematic equations to find how long it takes for the bus to travel between the two gas stations. Using the formula for time, \(t = \frac{distance}{speed}\), we calculated that it takes 45 seconds according to both the bus's and gas stations' frames.

Kinematics allows us to predict the future position of moving objects or compare their motion at different times, which is a good way to grasp the fundamental nature of motion in a straight-line path.