Two wave pulses on a string approach one another at the time t = 0, as shown in the figure below, except that pulse 2 is inverted so that it is a downward deflection of the string rather than an upward deflection. Each pulse moves with a speed of 1.0 m/s. Make a careful sketch of the resultant wave at the times t = 1.0 s, 2.0 s, 2.5 s, 3.0 s, and 4.0 s, assuming that the superposition principle holds for these waves, and that the absolute value of the height of each pulse is 4 mm in the figure below. (Do this on paper. Your instructor may ask you to turn in these sketches.) Also, determine the value of the resultant wave at x = 4.1 m at these five moments in time.

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t = 1 s __________ mm
t = 2 s ___________mm
t = 2.5 s __________ mm
t = 3 s __________mm
t = 4 s ___________mm

Answers are supposed to be in millimeter. Much appreciated.

To determine the resultant wave at different times and the value of the wave at a specific location, you can use the principle of superposition and the concept of wave interference. Here's how you can approach this problem:

1. Analyze the given wave pulses: The pulses are moving towards each other at a speed of 1.0 m/s. Pulse 1 has an upward deflection, while Pulse 2 is an inverted downward deflection. Both pulses have a height of 4 mm.

2. Determine the wavelength: Since the pulses have a speed of 1.0 m/s and the time intervals given are in seconds, the wavelength (λ) can be calculated as the product of the speed and the time interval. In this case, λ = 1.0 m/s * t (where t is given in seconds).

3. Sketch the wave at each specified time: To determine the shape of the resultant wave at the specified times, you need to add the displacements of the two pulses at different points on the string.

- At t = 1.0 s: Since the pulses have a speed of 1.0 m/s, both pulses will be 1.0 m away from their original positions. Sketch the sum of the displacements at x = 1.0 m.

- At t = 2.0 s: Both pulses would have traveled 2.0 m away from their original positions. Sketch the sum of the displacements at x = 2.0 m.

- At t = 2.5 s: Pulse 1 would have traveled 2.5 m to the left, while Pulse 2 would have traveled 2.5 m to the right. Sketch the sum of the displacements at x = -2.5 m (to the left of the original position).

- At t = 3.0 s: Pulse 1 would have traveled 3.0 m to the left, while Pulse 2 would have traveled 3.0 m to the right. Sketch the sum of the displacements at x = -3.0 m.

- At t = 4.0 s: Both pulses would have traveled 4.0 m away from their original positions. Sketch the sum of the displacements at x = 4.0 m.

4. Determine the value of the resultant wave at x = 4.1 m for each of the specified times: To find the value of the resultant wave at x = 4.1 m, find the corresponding displacement at that position on the resultant wave you sketched in step 3 for each specified time.

- t = 1.0 s: Find the value of the resultant wave at x = 4.1 m on the sketch from step 3.

- t = 2.0 s: Find the value of the resultant wave at x = 4.1 m on the sketch from step 3.

- t = 2.5 s: Find the value of the resultant wave at x = 4.1 m on the sketch from step 3.

- t = 3.0 s: Find the value of the resultant wave at x = 4.1 m on the sketch from step 3.

- t = 4.0 s: Find the value of the resultant wave at x = 4.1 m on the sketch from step 3.

Remember to use the principle of superposition to add the displacements of the two waves at each point in time.

To determine the shape of the resultant wave at each given time and its value at x = 4.1 m, we can follow these steps:

1. Draw a sketch of the initial wave pulses at t = 0 according to the given information.

2. Determine the positions of the pulses at the specified times by considering their respective speeds of 1.0 m/s.

3. Combine the two pulses using the superposition principle to obtain the resultant wave.

4. Find the value of the resultant wave at x = 4.1 m for each time.

Now, let's go step by step:

1. Draw a sketch of the initial wave pulses at t = 0, as shown in the given figure. Each pulse has a height of 4 mm.

2. To determine the positions of the pulses at the specified times, we will use the fact that each pulse moves with a speed of 1.0 m/s.

- At t = 1.0 s, pulse 1 moves to the right by 1.0 m, while pulse 2 (which is inverted) moves to the left by 1.0 m.
- At t = 2.0 s, pulse 1 moves to the right by an additional 1.0 m, reaching a total distance of 2.0 m, while pulse 2 moves to the left by an additional 1.0 m, reaching a total distance of 2.0 m.
- At t = 2.5 s, pulse 1 moves to the right by an additional 0.5 m, reaching a total distance of 2.5 m, while pulse 2 moves to the left by an additional 0.5 m, reaching a total distance of 2.5 m.
- At t = 3.0 s, pulse 1 moves to the right by an additional 1.0 m, reaching a total distance of 3.0 m, while pulse 2 moves to the left by an additional 1.0 m, reaching a total distance of 3.0 m.
- At t = 4.0 s, pulse 1 moves to the right by an additional 1.0 m, reaching a total distance of 4.0 m, while pulse 2 moves to the left by an additional 1.0 m, reaching a total distance of 4.0 m.

3. Combine the two pulses using the superposition principle to obtain the resultant wave.

- At each given time, plot the wave shape resulting from the superposition of the two pulses. The resultant wave will have alternating constructive and destructive interference regions.

4. Find the value of the resultant wave at x = 4.1 m for each time.

- At each given time, determine the position of x = 4.1 m on the resultant wave plot.
- Read the height of the resultant wave at x = 4.1 m to find the value.

Execute these steps on paper to obtain the sketches and values requested.

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