Question

Two football players move in a straight line directly toward one another. Their equations of motion are as follows: $$ \begin{aligned} &x_{1}=0.1 \mathrm{~m}+(-3.1 \mathrm{~m} / \mathrm{s}) t \\ &x_{2}=-6.3 \mathrm{~m}+(2.8 \mathrm{~m} / \mathrm{s}) t \end{aligned} $$ (a) Which player is moving faster? (b) At what time do the players collide?

Short answer

Player 1 is moving faster. They collide at approximately 1.08 seconds.

Step-by-step solution

Understanding the Players' Velocities

For player 1, the velocity \( v_1 \) is given as \(-3.1\ \text{m/s}\). Whereas for player 2, the velocity \( v_2 \) is \(2.8\ \text{m/s}\). The speeds are the absolute values of these velocities: \(|v_1|=3.1\ \text{m/s}\) and \(|v_2|=2.8\ \text{m/s}\).

Determine Which Player is Faster

Comparing the absolute values of the velocities, \(|v_1|=3.1\ \text{m/s}\) is greater than \(|v_2|=2.8\ \text{m/s}\). Thus, player 1 is moving faster than player 2.

Set Up the Collision Equation

To find the time when they collide, set the position equations equal: \(0.1 - 3.1t = -6.3 + 2.8t\). This equality occurs when the positions \(x_1\) and \(x_2\) are the same.

Solve for the Collision Time

Rearrange the equation: \(0.1 + 6.3 = 3.1t + 2.8t\). Combine terms: \(6.4 = 5.9t\). Solve for \(t\) to find \(t = \frac{6.4}{5.9}\).

Calculate the Exact Collision Time

Perform the division: \(t = \frac{6.4}{5.9} \approx 1.08\ \text{seconds}\). Therefore, the players collide approximately 1.08 seconds after starting from rest.

Key concepts

Velocity

Velocity is a vector quantity that describes the rate of change of an object's position. It's crucial to differentiate between speed and velocity:

• **Velocity** includes both the **speed** and the **direction** of the object's motion.
• **Speed** is simply the magnitude of velocity, which means it indicates how fast an object is moving without regard to direction.

In our exercise, we determine velocity by examining the coefficients of the time variable in the equations of motion.
For player 1: \[ v_1 = -3.1 \ ext{ m/s}\]The negative sign suggests movement in the opposite direction from the positive x-axis.
For player 2: \[ v_2 = 2.8 \ ext{ m/s}\]Here, the direction aligns with the positive x-axis. Remember, when comparing the speed (magnitude of velocity), we take the absolute values: \[ |v_1| = 3.1 \ ext{ m/s}\] and \[ |v_2| = 2.8 \ ext{ m/s}\].
This helps us see that player 1 is traveling faster than player 2, as speed only considers how fast something moves regardless of direction.

Equations of Motion

Equations of motion describe the mathematical relationship between an object's position, velocity, and time. They are fundamental to solving kinematics problems. In the presented exercise, the equations are:

• Player 1: \[ x_1 = 0.1 - 3.1t \]
• Player 2: \[ x_2 = -6.3 + 2.8t \]

These equations are in the form of linear equations representing constant velocity motion, ensuring direct, straight-line paths for both players.
In essence, when solving these equations, you want to pinpoint when their positions (\( x_1 \) and \( x_2 \)) are equal. That is when the collision occurs. Solving \[ 0.1 - 3.1t = -6.3 + 2.8t \]gives us the exact time of collision. Understanding these equations is key to predicting and determining motion outcomes, such as colliding positions and times in linear paths.

Collision

A collision occurs when two objects come into contact at a single point as their positions become the same at a certain time. In the realm of physics, determining the timing of a collision can involve solving simultaneous equations that describe each object's motion:

• The collision point in this context requires aligning position equations:
• \[ x_1 = x_2 \]

For our footballers, solving for that crucial moment meant equating
\[ 0.1 - 3.1t = -6.3 + 2.8t \] resulted in finding the collision time.
Completing the rearrangement and solving returned:\[ t \approx 1.08 \text{ seconds} \]
The players fuse their paths roughly 1.08 seconds into the game upon their simultaneous trajectories. Understanding such interactions is vital for grasping real-world applications of constant velocity motion and collisions.

Linear Motion

Linear motion references motion along a straight path, where the direction vector doesn't change. This motion is constant, with no rotations or deviations in path direction.
To conceptualize this, consider the football players each running on a straight line toward one another in the exercise:

• For linear motion, speed and velocity might remain constant.
• Both players maintain their magnitudes (3.1 m/s and 2.8 m/s, respectively).

The representations in the motion equations:\[ x_1 = 0.1 - 3.1t \]and\[ x_2 = -6.3 + 2.8t \]suit linear motion principles, where only speed and a straight path define the movement.
Linear motion simplifies our analysis of how objects travel and interact reliably, avoiding the complexities added by curves or angles.