Question

Object 1 starts at \(5.4 \mathrm{~m}\) and moves with a velocity of \(1.3 \mathrm{~m} / \mathrm{s}\). Object 2 starts at \(8.1 \mathrm{~m}\) and moves with a velocity of \(-2.2 \mathrm{~m} / \mathrm{s}\). The two objects are moving directly toward one another. (a) At what time do the objects collide? (b) What is the position of the objects when they collide?

Short answer

(a) About 0.77 seconds. (b) At approximately 6.4 meters.

Step-by-step solution

Define the position functions

The position of each object as a function of time can be written as follows. For Object 1: \[ x_1(t) = 5.4 + 1.3t \]For Object 2: \[ x_2(t) = 8.1 - 2.2t \]Where \( x_1 \) and \( x_2 \) are the positions of Object 1 and Object 2 respectively, and \( t \) is time in seconds.

Set the positions equal

To find out when the objects collide, set their position functions equal: \[ 5.4 + 1.3t = 8.1 - 2.2t \] This represents the point in time where both objects are at the same position.

Solve for time

Rearrange the equation from Step 2 to solve for \( t \): \[ 5.4 + 1.3t = 8.1 - 2.2t \]\[ 1.3t + 2.2t = 8.1 - 5.4 \]\[ 3.5t = 2.7 \]\[ t = \frac{2.7}{3.5} \]\[ t = 0.7714 \text{ seconds (approximately)} \] This is the time when the objects collide.

Find the position at collision

Substitute \( t = 0.7714 \) back into either position function (since both objects are at the same position then).Using Object 1's position formula: \[ x_1(0.7714) = 5.4 + 1.3 \times 0.7714 \]\[ x_1(0.7714) = 5.4 + 1.0028 \]\[ x_1(0.7714) \approx 6.4028 \text{ meters} \]Thus, the collision occurs at approximately 6.4 meters.

Key concepts

Kinematics

Kinematics in physics is all about the motion of objects without taking into account the forces that cause this motion. It's like describing an object's journey just by looking at how fast it's going and where it starts. In our example, we are studying two objects moving in a straight line. Kinematics helps us understand how their velocities affect their positions over time. Simple mathematical expressions called equations of motion help us predict where and when these objects will meet. This field is crucial as it forms the basis for more complex physics concepts, allowing you to predict various outcomes under different initial conditions. It paints a lively picture of moving objects using just numbers and variables!

Position Functions

Position functions describe where an object is located along a path as a function of time. In the given exercise, two position functions are used to track two objects:

• Object 1: \( x_1(t) = 5.4 + 1.3t \)
• Object 2: \( x_2(t) = 8.1 - 2.2t \)

The numbers in these functions allow us to calculate where each object is at any point in time. The initial terms \(5.4\) and \(8.1\) represent the starting positions, while the coefficients of \(t\) (such as \(1.3\) and \(-2.2\)) indicate their velocities. Total position at any given time is a combination of where it started and how much it moved.

Solving Equations

Solving equations is like piecing together a puzzle to find an unknown variable that makes the entire mathematical statement true. Here, setting the position functions equal finds when both objects are in the same place: \[5.4 + 1.3t = 8.1 - 2.2t\]To solve, add or subtract terms step by step to isolate the variable \(t\) on one side of the equation. By doing basic arithmetic operations, it becomes clear that \[t = 0.7714 \]This means at approximately 0.7714 seconds, both objects align perfectly, signifying a collision. Solving equations like this one is fundamental for predicting interactions between moving objects based on their paths.

Velocity

Velocity describes the speed of an object in a specific direction. In this exercise, Object 1 moves forward at \(1.3\) meters per second, while Object 2 moves backward at \(2.2\) meters per second. These velocities directly impact their position functions by changing their positions over time. Positive velocity means moving in the original direction, while negative velocity implies moving in the opposite direction. Understanding velocity is crucial for determining how fast positions change, ultimately helping predict when objects will meet on their paths.

Time of Collision

Time of collision is the exact moment when two objects meet each other along their paths. In this scenario, we determine that time by setting their position functions equal and solving for \(t\). The computed time of 0.7714 seconds is the point at which both objects occupy the same spot. By substituting back into the position functions, you can also confirm the physical location of the collision, which is around 6.4 meters from the start point of Object 1. Understanding collision time is essential for analyzing encounters between objects and ensuring safety and accuracy in real-world applications.