To solve this kinematics problem, we need to analyze the motion of both the receiver and the defender. We know that the receiver is running at a constant speed of 7 m/s, and we need to determine the speed at which the defender must run to intercept the receiver, as well as the distances traveled by both players. Let's break this down step by step.
Understanding the Motion
In this scenario, we have two players moving in a football game: the receiver and the defender. The receiver is moving in a straight line at a speed of 7 m/s, while the defender is running at an angle to intercept him. The key to solving this problem lies in understanding the relationship between their speeds and the distances they travel.
Setting Up the Problem
Let’s denote the following variables:
- v_r = speed of the receiver = 7 m/s
- v_d = speed of the defender (unknown)
- s = initial distance between the two players
- θ = angle of the defender's trajectory
Since both players are in uniform motion, we can use the equations of motion to find the relationship between their speeds and the distances they travel until interception.
Distance and Time Relationships
Let’s denote the time it takes for the defender to intercept the receiver as t. During this time, the distances traveled by both players can be expressed as:
- d1 (distance traveled by the receiver) = v_r * t = 7t
- d2 (distance traveled by the defender) = v_d * t
Since the defender is running at an angle, we can relate the distances using trigonometry. The horizontal component of the defender's distance must equal the distance the receiver travels, which means:
Establishing the Interception Condition
For the defender to intercept the receiver, the horizontal distance covered by the defender must equal the distance covered by the receiver. Therefore, we can set up the equation:
d2 * cos(θ) = d1
Substituting the expressions for d1 and d2 gives us:
v_d * t * cos(θ) = 7t
We can simplify this equation by canceling out t (assuming t is not zero):
v_d * cos(θ) = 7
Calculating the Defender's Speed
From the equation above, we can solve for the defender's speed:
v_d = 7 / cos(θ)
This equation tells us that the speed of the defender depends on the angle θ. The closer θ is to 0 degrees (running directly towards the receiver), the less speed the defender needs. Conversely, as θ increases, the defender must run faster to intercept the receiver.
Finding Distances Traveled
Now that we have the defender's speed, we can find the distances traveled by both players:
Using the time t again, we can express the distances:
- d1 = 7t
- d2 = v_d * t = (7 / cos(θ)) * t
To find the exact distances, we would need the time t or the initial distance s between the players. If we know s, we can relate it to the distances traveled by both players using the Pythagorean theorem, especially if the angle θ is known.
Conclusion
In summary, the speed at which the defender must run to intercept the receiver is given by v_d = 7 / cos(θ). The distances traveled by the receiver and the defender can be calculated once we have the time of travel or the initial distance between them. This approach illustrates the interplay between speed, distance, and angle in kinematics, providing a clear path to solving similar problems in the future.
