10 In a football game the defender is running to intercept the receiver. It is given that this is a kinematics problem in which both players are experiencing uniform motion. The receiver is running at 7 m/s. The blue and green dots represent the initial positions of the players. The angle ? represents the running trajectory of the defender, as shown, and s is the initial distance between the two players. At what velocity must the defender run at in order to intercept the receiver, as shown in the figure below? What is the distance traveled by the receiver (d1) and the defender (d2)?

To solve this problem, we need to analyze the motion of both the receiver and the defender using the principles of kinematics. Since both players are moving at uniform speeds, we can apply the concepts of relative motion and geometry to determine the defender's required velocity and the distances traveled by both players.

Understanding the Scenario

We have two players: the receiver running at a constant speed of 7 m/s and the defender who needs to intercept him. The initial distance between them is represented as 's', and the angle of the defender's trajectory is denoted as ?. The goal is to find the velocity of the defender (let's call it Vd) required to intercept the receiver, as well as the distances they each travel during the interception.

Setting Up the Problem

To find the velocity of the defender, we can use the following steps:

• Define the time it takes for the receiver to cover a certain distance.
• Use the geometry of the situation to relate the distances traveled by both players.
• Set up equations based on their speeds and the time taken to meet.

Calculating Time and Distances

Let’s denote the time it takes for the receiver to travel a distance 'd1' as 't'. Since the receiver runs at 7 m/s, we can express the distance traveled by the receiver as:

d1 = 7t

For the defender, who runs at velocity Vd at an angle ?, the distance traveled can be expressed as:

d2 = Vd * t

Using Geometry

Since the defender is running at an angle, we can relate the distances using trigonometry. The horizontal component of the defender's distance can be expressed as:

d2 * cos(?) = s + d1 * cos(θ)

Here, θ is the angle at which the receiver is moving relative to the horizontal. This equation helps us relate the distances traveled by both players.

Setting Up the Equations

Now, we can substitute d1 into the equation:

d2 * cos(?) = s + (7t) * cos(θ)

Substituting d2 from our earlier expression gives:

Vd * t * cos(?) = s + (7t) * cos(θ)

Solving for the Defender's Velocity

We can rearrange this equation to isolate Vd:

Vd = (s + 7t * cos(θ)) / (t * cos(?))

To find the exact value of Vd, we need to know the angle ? and the initial distance 's'. Once we have those values, we can calculate the time 't' it takes for both players to meet, allowing us to find the defender's velocity.

Calculating Distances Traveled

Once we have Vd, we can easily calculate d1 and d2:

• d1 = 7t
• d2 = Vd * t

By substituting the value of 't' from the earlier equations, we can find the distances traveled by both players during the interception.

Conclusion

In summary, to determine the velocity of the defender and the distances traveled by both players, we need to set up the equations based on their speeds and the geometry of their paths. Once we have the necessary values, we can solve for the unknowns systematically. If you have specific values for 's' and the angle ?, we can plug those in to get numerical answers!