Question

You are riding in a car on a straight stretch of a two-lane highway with a speed of \(26 \mathrm{~m} / \mathrm{s}\). At a certain time, which we will choose to be \(t=0\), you notice a truck moving toward you in the other lane. The truck has a speed of \(31 \mathrm{~m} / \mathrm{s}\) and is \(420 \mathrm{~m}\) away at \(t=0\). (a) Write the position-time equations of motion for your car and for the truck in the other lane. (b) Plot the two equations of motion on a position-time graph. (c) At what time do you and the truck pass one another, going in opposite directions?

Short answer

The cars pass each other at approximately 7.37 seconds.

Step-by-step solution

Define Reference Points and Directions

Since you're moving towards the truck, we'll choose your starting position as the origin. Both you and the truck are moving towards each other, so your movement is considered in the positive direction, while the truck's movement is in the negative direction along the x-axis.

Determine Initial Velocities and Positions

Your car travels at a velocity of \(v_{car} = 26 \, \mathrm{m/s}\) and starts from the origin \(x_{car}(0) = 0\). The truck travels at \(v_{truck} = -31 \, \mathrm{m/s}\) (negative because it's moving towards you) and starts \(420 \, \mathrm{m}\) away at \(x_{truck}(0) = 420 \, \mathrm{m}\).

Write Position-Time Equations

Using the equation \(x(t) = x_0 + v t\), write the equations for your car and the truck:- Car: \(x_{car}(t) = 0 + 26t = 26t\)- Truck: \(x_{truck}(t) = 420 - 31t\)

Plot Position-Time Graphs

To graph these equations, plot time on the x-axis and position on the y-axis. Plot the line for the car with slope 26 and initial position 0. Plot the line for the truck with slope -31 and initial position 420.

Find the Intersection Point

The two vehicles pass each other when their positions are equal, i.e., \(x_{car}(t) = x_{truck}(t)\). Solve the equation \(26t = 420 - 31t\) for \(t\).

Solve for Time

Rearrange the equation to: \[ 26t + 31t = 420 \]\[ 57t = 420 \]\[ t = \frac{420}{57} \approx 7.37 \, \mathrm{s}\]

Confirm the Solution

Substitute \(t = 7.37 \, \mathrm{s}\) back into both position equations to ensure they equate to the same value. Both should give \(x \approx 192.62 \, \mathrm{m}\), confirming that the cars pass each other at \(t = 7.37 \, \mathrm{s}\).

Key concepts

Position-Time Graphs

A position-time graph is a fundamental concept used to visualize and interpret the motion of objects over time. In such a graph, time is typically plotted on the x-axis, while position is plotted on the y-axis.
These graphs allow us to easily determine the speed and direction of moving objects based on the slope of the line.

• A positive slope indicates movement in the forward direction, while a negative slope signifies reverse motion.
• The steepness of the slope demonstrates the speed: a steeper slope means higher velocity.

By plotting different objects on the same graph, you can easily compare their speeds and positions over time. For our scenario, plotting the car and the truck as two separate lines will show their motion toward each other. The car, moving at 26 m/s, will have a line with a positive slope, while the truck, moving toward the car at 31 m/s, will show a line with a negative slope. This visualization helps in identifying the point at which they will meet.

Equations of Motion

The equations of motion are mathematical formulas used to describe the position of an object as a function of time, considering its initial position and velocity. They are crucial in predicting where an object will be at any given time.

• The general form of the position-time equation is given as: \( x(t) = x_0 + v t \), where \( x(t) \) is the position at time \( t \), \( x_0 \) is the initial position, and \( v \) is the velocity.

Applying this equation to our exercise:

• For the car, starting at the origin with a speed of 26 m/s, the equation becomes \( x_{car}(t) = 26t \).
• For the truck, starting 420 meters away and moving toward the car, the equation is \( x_{truck}(t) = 420 - 31t \).

These equations elucidate how positions of the car and truck change over time, serving as crucial tools for predicting collisions or meeting points when plotted or equated.

Intersection Point

The intersection point of two position-time lines on a graph represents the time and position at which two moving objects meet. In calculations, this is where the equations of motion for the objects equate to each other, meaning both objects occupy the same position at the same moment.
For the car and the truck:

• We set their position equations equal: \( 26t = 420 - 31t \).
• Solving this yields \( 57t = 420 \), from which \( t = \frac{420}{57} \approx 7.37 \) seconds, indicating the time of their meeting.

At approximately 7.37 seconds, both vehicles reach the same position along their trajectories, meaning they pass each other on the road. This calculation can be confirmed by substituting the time back into their position equations, both of which should result in an equivalent position value around 192.62 meters, ensuring that the intersection point is accurate.