Chapter 2: Problem 100
Two football players move in a straight line directly toward one another.
Their equations of motion are as follows:
Short Answer
Expert verified
Player 1 is moving faster. They collide at approximately 1.08 seconds.
Step by step solution
01
Understanding the Players' Velocities
For player 1, the velocity is given as . Whereas for player 2, the velocity is . The speeds are the absolute values of these velocities: and .
02
Determine Which Player is Faster
Comparing the absolute values of the velocities, is greater than . Thus, player 1 is moving faster than player 2.
03
Set Up the Collision Equation
To find the time when they collide, set the position equations equal: . This equality occurs when the positions and are the same.
04
Solve for the Collision Time
Rearrange the equation: . Combine terms: . Solve for to find .
05
Calculate the Exact Collision Time
Perform the division: . Therefore, the players collide approximately 1.08 seconds after starting from rest.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
For player 1: The negative sign suggests movement in the opposite direction from the positive x-axis.
For player 2: Here, the direction aligns with the positive x-axis. Remember, when comparing the speed (magnitude of velocity), we take the absolute values: and .
This helps us see that player 1 is traveling faster than player 2, as speed only considers how fast something moves regardless of direction.
- **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.
For player 1:
For player 2:
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:
In essence, when solving these equations, you want to pinpoint when their positions ( and ) are equal. That is when the collision occurs. Solving 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.
- Player 1:
- Player 2:
In essence, when solving these equations, you want to pinpoint when their positions (
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:
resulted in finding the collision time.
Completing the rearrangement and solving returned:
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.
- The collision point in this context requires aligning position equations:
Completing the rearrangement and solving returned:
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:
and 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.
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).
Linear motion simplifies our analysis of how objects travel and interact reliably, avoiding the complexities added by curves or angles.