Two pulses are started on a stretched string. An upward pulse is started on the

Question

Two pulses are started on a stretched string. An upward pulse is started on the right that moves to the left. At the same instant, a downward pulse is started on the left that moves to the right. At t = 0 their peaks are separated by a distance of 4 cm as shown below. The pulses move on the string with a speed 200 cm/s.

The figure below shows some graphs that could represent properties of these pulses on the string. The vertical scales in the graphs are arbitrary and not necessarily the same. The horizontal scales all represent positions on the string (x).

For the situation and the properties (a) - (f) below, select which graph provides the best representation of the given property. If none of the graphs are correct, write "N".

Which graph best represents the appearance of the string at time t = 0.01 s?  
Which graph best represents the appearance of the string at a time t = 0.02 s?  
Which graph best represents the y-component of the velocity of the bits of the string at a time t = 0?  
Which graph best represents the y-component of the velocity of the bits of the string at a time t = 0.01 s?  
Which graph best represents the y-component of the velocity of the bits of the string at a time t = 0.02 s?  
Which graph best represents the x-component of the velocity of the bits of the string at a time t = 0?

Explanation / Answer

1. Graph E best represents the appearance of the string at time t = 0.01 s.

(As speed = 200cm/s, distance travelled by pulse in time 0.01s is equal to 200cm/s * 0.01s = 2 cm. So the both pulses will move further by 2cm. Upward pulse will be centered at 0 & Downward pulse will also be centered at 0, such that they will cancel each other. Hence the string will be seen as in graph E)

2. Graph B best represents the appearance of the string at time t = 0.02 s

(As speed = 200cm/s, distance travelled by pulse in time 0.02s is equal to 200cm/s * 0.01s = 4 cm. So the both pulses will move further by 4cm. Upward pulse will be centered at +2 & Downward pulse will be centered at -2, hence the string will be seen as in graph B)

3. Graph C best represents the y-component of the velocity of the bits of the string at a time t = 0

(For velocity graph, we shall consider the slope or gradient of the curve at time 0s.

For upward pulse as we go from time -3s towards -1s, slope is increasing from 0 to a maximum value, then again reducing to 0 at time -2s, then slope is becoming negative, attaining a maximum value in negative direction & again becoming 0 at time -1s. Hence the first curve in graph C is the best representation.

For downward pulse, as we go from time 1s towards 3s, slope is decreasing from 0 to a maximum value in negative direction & then becoming 0 again at time 2s. Slope is then increasing from 0 to a maximum value, then again reducing to 0 at time 3s. Hence the second curve in graph C is the best representation.)

4.  Graph E best represents the y-component of the velocity of the bits of the string at a time t = 0.01 s.

( As discussed in part 3 above, the shape of the y-component of the velocity of upward pulse & downward pulse will be same. But at time 0.01s, the velocity of upward pulse will shift by +2cm centered at 0 & the velocity of downward pulse will shift by -2cm centered at 0. They will cancel out each other & so the Graph E will be the best representation)

5. Graph D best represents the y-component of the velocity of the bits of the string at a time t = 0.02 s.

(As Graph B best represents the appearance of the string at time t = 0.02 s, which is exactly the inverse of appearance of the string at time t = 0 s. So the graph for the y-component of the velocity of the bits of the string at a time t = 0.02 s should be exactly opposite of Graph C, which represents the y-component of the velocity of the bits of the string at a time t = 0. It can also be solved by taking slope of Graph B as explained in part 3)

6. Graph E best represents the x-component of the velocity of the bits of the string at a time t = 0

(Bits of String are not moving in x-direction, so its velocity in x-direction is always 0, hence Graph E whose value is 0 at all the time)

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