A pinball bounced around its machine before coming to rest between two bumpers. Before coming to rest, the ball's displacement was recorded as a series of vectors. All the angles are measured counterclockwise from the horizontal x axis.

d1: 83.0 cm at 90.0°

d2: 57.0 cm at 153°

d3: 67.0 cm at 227°

Determine the magnitude dtotal and direction 0total (measured counterclockwise from the +x axis) of the ball's total displacement dtotal

To find the total displacement of the pinball, we need to add the given displacement vectors \(\vec{d_1}\), \(\vec{d_2}\), and \(\vec{d_3}\) using their components.

**Step 1: Break each vector into its components.**

1. **Vector \(\vec{d_1}\):**  
   - Magnitude: \(83.0 \, \text{cm}\)  
   - Angle: \(90.0^\circ\)  
   - Components:
     \[
     \vec{d_1} = (83.0 \cos 90^\circ, 83.0 \sin 90^\circ) = (0, 83.0)
     \]

2. **Vector \(\vec{d_2}\):**  
   - Magnitude: \(57.0 \, \text{cm}\)  
   - Angle: \(153^\circ\)  
   - Components:
     \[
     \vec{d_2} = (57.0 \cos 153^\circ, 57.0 \sin 153^\circ)
     \]

3. **Vector \(\vec{d_3}\):**  
   - Magnitude: \(67.0 \, \text{cm}\)  
   - Angle: \(227^\circ\)  
   - Components:
     \[
     \vec{d_3} = (67.0 \cos 227^\circ, 67.0 \sin 227^\circ)
     \]

**Step 2: Calculate the components for \(\vec{d_2}\) and \(\vec{d_3}\).**

Now, let's calculate the components for \(\vec{d_2}\) and \(\vec{d_3}\) and sum them up to get the total components.

**Step 3: Sum the x and y components to get the total displacement vector.**

**Step 4: Calculate the magnitude and direction of the total displacement vector.**

Let's go through these steps:The total displacement of the pinball, \(\vec{d}_{\text{total}}\), has a magnitude of approximately **113.6 cm** and a direction of approximately **148.2°** measured counterclockwise from the positive x-axis.

Would you like more details or have any questions about the calculation? Here are some related questions you might find interesting:

1. How would the result change if one of the angles were different?
2. What is the effect of changing the magnitude of one of the vectors on the total displacement?
3. How can we visualize these vectors on a coordinate plane?
4. What would happen if the vectors were in three dimensions instead of two?
5. How do you convert between Cartesian and polar coordinates for vectors?

**Tip:** When adding vectors, always break them into their x and y components. This makes it easy to sum them up and find the total displacement or resultant vector.

A pinball bounced around its machine before coming to rest between two bumpers. Before coming to rest, the ball's displacement was recorded as a series of vectors. All the angles are measured counterclockwise from the horizontal x axis.
d1: 83.0 cm at 90.0°
d2: 57.0 cm at 153°
d3: 67.0 cm at 227°
Determine the magnitude dtotal and direction 0total (measured counterclockwise from the +x axis) of the ball's total displacement dtotal.

Learn how to determine the total displacement of a pinball in motion by adding three displacement vectors: 83.0 cm at 90°, 57.0 cm at 153°, and 67.0 cm at 227°. This problem uses vector addition and trigonometry concepts, including calculating magnitude and direction of the resultant displacement. Suitable for high school students.

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