bill steps off a 3.0-m-high diving board and drops to the water below. At the same time Ted jumps upward with a speed of 4.2 m/s from a 1.0-m-high diving board. choosing the origin to be at the water's surface and upward to be the positive direction, write x-versus-t equations of motion for both bill and ted.

To write the x-versus-t equations of motion for both Bill and Ted, we need to consider their vertical motions separately.

1. For Bill, who steps off a 3.0 m-high diving board and drops to the water below:
- Initial position (x₀) = 3.0 m (above the water's surface)
- Initial velocity (v₀) = 0 m/s (since Bill starts from rest)
- Acceleration (a) = -9.8 m/s² (acceleration due to gravity acting downward)

The x-versus-t equation for Bill's motion can be given by:
x(t) = x₀ + v₀t + (1/2)at²

Substituting the values, the equation becomes:
x(t) = 3.0 m + 0 m/s * t + (1/2)(-9.8 m/s²)t²
x(t) = 3.0 m - 4.9 m/s²t²

2. For Ted, who jumps upward with a speed of 4.2 m/s from a 1.0 m-high diving board:
- Initial position (x₀) = -1.0 m (below the water's surface)
- Initial velocity (v₀) = 4.2 m/s (upward)
- Acceleration (a) = -9.8 m/s² (acceleration due to gravity acting downward)

The x-versus-t equation for Ted's motion can be given by:
x(t) = x₀ + v₀t + (1/2)at²

Substituting the values, the equation becomes:
x(t) = -1.0 m + (4.2 m/s)t + (1/2)(-9.8 m/s²)t²
x(t) = -1.0 m + 4.2 m/s * t - 4.9 m/s²t²

Therefore, the x-versus-t equations of motion for Bill and Ted are:
Bill: x(t) = 3.0 m - 4.9 m/s²t²
Ted: x(t) = -1.0 m + 4.2 m/s * t - 4.9 m/s²t²

To write the x-versus-t equations of motion for Bill and Ted, we need to use the equation of motion:

\[x = x_0 + v_0t + \frac{1}{2}at^2\]

where:
- \(x\) is the position at time \(t\)
- \(x_0\) is the initial position
- \(v_0\) is the initial velocity
- \(a\) is the acceleration
- \(t\) is the time

For Bill:
- \(x_0\) is 3.0 m (height of the diving board)
- \(v_0\) is 0 m/s (Bill is initially at rest)
- \(a\) is the acceleration due to gravity, which is approximately \(9.8 \, \text{m/s}^2\) (acceleration towards the ground is positive)

Therefore, the x-versus-t equation for Bill is:
\[x_{\text{Bill}} = 3.0 - \frac{1}{2} \times 9.8 \times t^2\]

For Ted:
- \(x_0\) is 1.0 m (height of the diving board)
- \(v_0\) is 4.2 m/s (Ted jumps upward with a positive velocity)
- \(a\) is the acceleration due to gravity, which is approximately \(9.8 \, \text{m/s}^2\) (acceleration towards the ground is positive)

Therefore, the x-versus-t equation for Ted is:
\[x_{\text{Ted}} = 1.0 + 4.2t - \frac{1}{2} \times 9.8 \times t^2\]

These equations describe the height of Bill and Ted at any given time \(t\) after they jump.

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