Question

Object 1 starts at \(25 \mathrm{~m}\) and moves with a velocity of \(-5.6 \mathrm{~m} / \mathrm{s}\). Object 2 starts at \(13 \mathrm{~m}\) and moves directly toward object 1 . The two objects collide \(0.61 \mathrm{~s}\) after starting. (a) What is the velocity of object 2? (b) What is the position of the objects when they collide?

Short answer

(a) The velocity of Object 2 is approximately 14.076 m/s. (b) The position at collision is approximately 21.584 m.

Step-by-step solution

Understand the problem

We are given two objects: Object 1 is moving with a velocity of \(-5.6 \mathrm{~m/s}\) from \(25 \mathrm{~m}\), and Object 2 starts at \(13 \mathrm{~m}\). We need to determine the velocity of Object 2 such that the objects collide after \(0.61 \mathrm{~s}\). We also need to find their position at the point of collision.

Calculate the position of Object 1 at collision

The position of Object 1 can be calculated using the formula for constant velocity motion: \[\text{Position of Object 1} = \text{Initial Position} + \text{Velocity} \times \text{Time}\]Given, Initial Position = \(25 \mathrm{~m}\), Velocity = \(-5.6 \mathrm{~m/s}\), and Time = \(0.61 \mathrm{~s}\):\[\text{Position of Object 1} = 25 + (-5.6) \times 0.61 = 25 - 3.416 = 21.584 \mathrm{~m}\]

Express the position of Object 2 at collision

Let \(v\) be the velocity of Object 2. Its position can also be calculated using constant velocity motion:\[\text{Position of Object 2} = \text{Initial Position} + \text{Velocity} \times \text{Time}\]Using Initial Position = \(13 \mathrm{~m}\) and Time = \(0.61 \mathrm{~s}\):\[\text{Position of Object 2} = 13 + v \times 0.61\]

Establish collision condition

For the two objects to collide, their positions must be equal at \(0.61 \mathrm{~s}\). Therefore, set the position expressions equal:\[21.584 = 13 + 0.61v\]

Solve for the velocity of Object 2

From the equation, solve for \(v\):\[21.584 = 13 + 0.61v\]First, subtract 13 from both sides:\[8.584 = 0.61v\]Then, divide both sides by 0.61:\[v = \frac{8.584}{0.61} \approx 14.076 \mathrm{~m/s}\]Thus, the velocity of Object 2 is approximately \(14.076 \mathrm{~m/s}\).

Confirm the position at collision

Using either the position of Object 1 or Object 2 at collision (they should be equal), we see they both calculate to approximately \(21.584 \mathrm{~m}\). Thus, the collision position is:\[21.584 \mathrm{~m}\]

Key concepts

Collision

In kinematics, a collision occurs when two or more objects come into contact or influence each other, generally along their path of motion. For a collision to happen, the positions of the objects involved must be the same at a certain point in time.
In our exercise, Object 1 and Object 2 collide after moving for a certain period. This fact allows us to set conditions where the positions of these objects are equal at the time of collision.
Detecting a collision is a fundamental problem in motion analysis, allowing us to learn more about object behavior before and after an interaction. It can involve calculating trajectories, velocities, and sometimes the effects post-collision depending on the objects' characteristics and the energy involved. But in this simple introductory step, we're learning how to predict the meeting point by equating their positions.

Constant Velocity Motion

Constant velocity motion is a straightforward type of motion where an object moves at a steady speed in a straight line. The velocity does not change over time, indicating no acceleration is involved, thus simplifying our position calculations.
When an object moves with constant velocity, its future position at any time can be calculated from its starting position plus the velocity multiplied by the time elapsed. The equation for this is:

• Position = Initial Position + (Velocity × Time)

In the given exercise, both Object 1 and Object 2 are in a state of constant velocity motion. The simplicity of this motion type allows easy calculation of how far each object travels in a given amount of time, like the 0.61 seconds in which both objects eventually collide.
Understanding constant velocity motion is crucial for more complex motion analysis, as it provides the groundwork for considering variable velocities and accelerations.

Position Calculation

Calculating the position of an object accurately is key when analyzing motions, especially to determine events like collisions. In the exercise, determining where the objects collided required knowing their position after moving at their respective velocities for a set time period.
For Object 1 starting at 25 m with a velocity of -5.6 m/s, its position at the time of collision was calculated with:

• Position = 25 + (-5.6 × 0.61)

leading to a final position of 21.584 m.
Meanwhile, Object 2 started at 13 m, and its velocity is calculated from the condition that its position needed to be equal to Object 1's position when they collided.
Knowing these calculations allows us to predict or verify static outcomes in motion analyses such as collision points, helping to understand and potentially simulate object behavior in physics.