To determine which displacement has the largest magnitude, let's go through the displacements for each given pair of initial and final coordinates step-by-step.
1. Option a: [tex]\(x_i = 4 \, \text{m}\)[/tex], [tex]\(x_f = 6 \, \text{m}\)[/tex]
- Displacement [tex]\( \Delta x_a = x_f - x_i = 6 \, \text{m} - 4 \, \text{m} = 2 \, \text{m} \)[/tex]
- Magnitude of displacement: [tex]\( |2 \, \text{m}| = 2 \, \text{m} \)[/tex]
2. Option b: [tex]\(x_i = -4 \, \text{m}\)[/tex], [tex]\(x_f = -8 \, \text{m}\)[/tex]
- Displacement [tex]\( \Delta x_b = x_f - x_i = -8 \, \text{m} - (-4 \, \text{m}) = -8 \, \text{m} + 4 \, \text{m} = -4 \, \text{m} \)[/tex]
- Magnitude of displacement: [tex]\( |-4 \, \text{m}| = 4 \, \text{m} \)[/tex]
3. Option c: [tex]\(x_i = -4 \, \text{m}\)[/tex], [tex]\(x_f = 2 \, \text{m}\)[/tex]
- Displacement [tex]\( \Delta x_c = x_f - x_i = 2 \, \text{m} - (-4 \, \text{m}) = 2 \, \text{m} + 4 \, \text{m} = 6 \, \text{m} \)[/tex]
- Magnitude of displacement: [tex]\( |6 \, \text{m}| = 6 \, \text{m} \)[/tex]
4. Option d: [tex]\(x_i = 4 \, \text{m}\)[/tex], [tex]\(x_f = -2 \, \text{m}\)[/tex]
- Displacement [tex]\( \Delta x_d = x_f - x_i = -2 \, \text{m} - 4 \, \text{m} = -2 \, \text{m} - 4 \, \text{m} = -6 \, \text{m} \)[/tex]
- Magnitude of displacement: [tex]\( |-6 \, \text{m}| = 6 \, \text{m} \)[/tex]
Now, let's compare the magnitudes of these displacements:
- Magnitude of displacement for option a: [tex]\( 2 \, \text{m} \)[/tex]
- Magnitude of displacement for option b: [tex]\( 4 \, \text{m} \)[/tex]
- Magnitude of displacement for option c: [tex]\( 6 \, \text{m} \)[/tex]
- Magnitude of displacement for option d: [tex]\( 6 \, \text{m} \)[/tex]
The largest magnitude of displacement is [tex]\( 6 \, \text{m} \)[/tex], which occurs in both option c and option d. Therefore, the displacements with the largest magnitude are given in:
Option c: [tex]\( x_i = -4 \, \text{m} \)[/tex], [tex]\( x_f = 2 \, \text{m} \)[/tex] and Option d: [tex]\( x_i = 4 \, \text{m} \)[/tex], [tex]\( x_f = -2 \, \text{m} \)[/tex].
