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

Velocity of Object 2: 14.08 m/s; Collision position: 21.584 m.

Step-by-step solution

Setup Initial Conditions

Object 1 begins at a position of $25$ m and moves with a velocity of $-5.6$ m/s. Object 2 begins at a position of $13$ m. Both objects are moving towards each other, and they collide after $0.61$ s.

Calculate Object 1's Position at Collision

Using the equation for motion, $x=x_0+vt$, where $x_0$ is the initial position, $v$ is the velocity, and $t$ is the time, we calculate Object 1’s position when they collide:$$x_1=25+(-5.6)\times 0.61$$Calculating gives:$x_1=25-3.416=21.584\text{ m}$.

Calculate Object 2's Position at Collision

At the moment of collision, both objects are at the same position. Therefore, the position of Object 2 at collision is $21.584$ m, which we calculated for Object 1.

Determine Object 2's Velocity

The displacement of Object 2 during the time interval is the difference between its initial position and its position at the time of collision:$$x_{extdisplacement}=21.584-13=8.584\text{ m}$$The velocity is then:$$v_2=\frac{x_{extdisplacement}}{t}=\frac{8.584}{0.61}\approx 14.08\text{ m/s}$$

Conclusion and Answer

Object 2's velocity is approximately $14.08$ m/s towards Object 1, and the objects collide at a position of $21.584$ m.

Key concepts

Motion Equation

The motion equation is a fundamental concept in physics that helps us predict the future position of an object in motion, based on its current state. The most commonly used motion equation is: $$x=x_0+vt$$ Here:

• $x$ represents the object's position at a specific time.
• $x_0$ is the initial position at the beginning of the motion.
• $v$ stands for velocity, the rate of change of position over time.
• $t$ is the time duration for which the object has been moving.

To use this equation effectively, you need to know the initial conditions, such as the starting position and velocity of the object. From there, you can calculate where the object will be after a certain amount of time, like in the case of Object 1, which started at 25 meters and moved with a velocity of -5.6 m/s. By plugging in these values along with the time to collision (0.61 s), we find that it will be at position 21.584 meters when the collision occurs.

Velocity Calculation

Velocity calculation is a crucial aspect when analyzing the motion of objects, particularly when they are moving towards each other like in collisions. Velocity is defined as the speed of an object in a specified direction, and it is calculated with the formula:$$v=\frac{\text{displacement}}{\text{time}}$$ For Object 2, we know:

• The initial position was 13 meters.
• The position at collision, shared with Object 1, is 21.584 meters.
• Time to collision was 0.61 seconds.

The displacement, which is the change in position, can be calculated by subtracting the initial position from the position at collision:$$x_{\text{displacement}}=21.584-13=8.584\text{ meters}$$With this displacement, we compute Object 2's velocity:$$v_2=\frac{8.584}{0.61}\approx 14.08\text{ m/s}$$This calculation tells us Object 2 moved towards Object 1 at approximately 14.08 meters per second to reach the collision point.

Position Calculation

Position calculation refers to determining the precise point in space an object occupies at a given time. In problems involving motion and collisions, knowing the exact position is key to understanding the interaction between objects.For our given problem:

• Initially, Object 1 is at 25 meters, and Object 2 at 13 meters.
• They move towards each other and collide after 0.61 seconds.

Using the motion equation for Object 1, the position at the time of collision was calculated:$$x_1=25+(-5.6)\times 0.61$$This simplifies to:$$x_1=21.584\text{ meters}$$Since both objects clearly collide, they share this position at the point of collision. Thus, the position calculation not only tracks where each object has gone but also confirms that their paths intersect. Understanding how to find these specific points helps visualize and solve many real-world motion scenarios.