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A police officer is driving due north at a constant speed of \(29 \mathrm{m} / \mathrm{s}\) relative to the ground when she notices a truck on an east-west highway ahead of her, driving west at high speed. She finds that the truck's speed relative to her car is \(48 \mathrm{m} / \mathrm{s}\) (about \(110 \mathrm{mph}\) ). (a) Sketch the vector triangle that shows how the truck's velocity relative to the ground is related to the police car's velocity relative to the ground and to the truck's velocity relative to the police car. The sketch need not be to scale, but the velocity vectors should be oriented correctly and bear the appropriate labels. (b) What is the truck's speed, relative to the ground?

Step by step solution

01

Understand the Problem

We have three velocities to consider: the police car's movement directly north at \(29\,\mathrm{m/s}\) relative to the ground, the truck's movement west relative to the police car at \(48 \, \mathrm{m/s}\), and the truck's movement relative to the ground, which we need to find.

02

Sketch the Vector Triangle

Draw a vector pointing upwards (north) to represent the police car's velocity of \(29\, \mathrm{m/s}\). From the tip of this vector, draw another vector to the left (west) with a magnitude of \(48\, \mathrm{m/s}\) to represent the truck's velocity relative to the police car. The resultant vector from the starting point of the first vector to the end point of the second vector represents the truck's velocity relative to the ground.

03

Apply the Pythagorean Theorem

Since the vectors form a right triangle, use the Pythagorean theorem to calculate the truck's velocity relative to the ground. Let \(v_t\) be the truck's velocity relative to the ground. According to the theorem, we have: \[ v_t^2 = (29\, \mathrm{m/s})^2 + (48\, \mathrm{m/s})^2 \]Calculate \(v_t\).

04

Calculate Magnitude

Calculate the magnitude of the truck’s speed relative to the ground:\[ v_t = \sqrt{(29^2 + 48^2)}\]This results in:\[ v_t = \sqrt{841 + 2304} = \sqrt{3145} = 56.08\, \mathrm{m/s} \]

05

Conclusion

The truck’s speed relative to the ground is approximately \(56.08\, \mathrm{m/s}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector Triangle in Relative Velocity

In the study of relative velocity, a vector triangle is a visual representation that helps us understand the relationship between multiple velocities.
For instance, in our exercise, we have the police car moving north at 29 m/s, and we need to determine the velocity of the truck relative to the ground.

Here's how you can think about it:

• The vector representing the police car points north, which is straight up when sketching.
• The truck's velocity relative to the police car, which is given as 48 m/s west, forms the second vector pointing leftwards.
• These two vectors create a right angle, hence forming a triangle.
• The third vector, representing the truck's velocity relative to the ground, completes the triangle.

This right-angled triangle helps us use mathematical tools like the Pythagorean Theorem to resolve these vectors into a quantitative understanding of velocities. Sketching this triangle aids in visualizing not just directions but also how these movements combine in a system.

Using the Pythagorean Theorem

The Pythagorean Theorem is a fundamental mathematical principle used when dealing with right-angled triangles. It states that for a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In terms of our vector triangle:

• The northward movement of the police car is one side, measured as 29 m/s.
• The westward movement of the truck relative to the car is the other side, measuring 48 m/s.
• The truck's velocity relative to the ground is the hypotenuse we need to find.

Using the Pythagorean Theorem: \[v_t^2 = (29)^2 + (48)^2\]By calculating this, we solve for the hypotenuse, which gives us the truck's speed relative to the ground. Breaking down these steps helps us find that the truck's velocity, measured as approximately 56.08 m/s, arises naturally from the geometric relationships in the triangle.

Calculate Truck's Velocity Relative to the Ground

Velocity calculation simplifies complex motion into understandable terms. Here, the goal is to find the truck's speed relative to the ground using the vector triangle and mathematical principles.

To calculate it:

• Start with the sides of the triangle: northward 29 m/s for the police car, and westward 48 m/s for the truck relative to the police car.
• Use the formula from the Pythagorean theorem: \[ v_t = \sqrt{(29^2 + 48^2)} \]
• Work through the calculation: \[ v_t = \sqrt{841 + 2304} = \sqrt{3145} = 56.08 \, \mathrm{m/s} \]

This results in the magnitude of the truck's speed, showing that careful application of these principles allows us to translate the abstract concept of relative motion into concrete numbers. Understanding this approach helps master problems involving vector addition and relative velocities effectively.