Question

A scuba diver makes a slow descent into the depths of the ocean. His vertical position with respect to a boat on the surface changes several times. He makes the first stop 8.0 m from the boat but has a problem with equalizing the pressure, so he ascends 5.0 m and then continues descending for another 12.0 m to the second stop. From there, he ascends 4.0 m and then descends for 14.0 m, ascends again for 8.0 m and descends again for 11.0 m, where he makes a stop, waiting for his buddy. Assuming the positive direction up to the surface (+ĵ), express his net vertical displacement vector (in m) in terms of the unit vector ĵ. (Express your answer in vector form.)

Answer 1

Answer:

The total vertical displacement is $\vec{D}_{diver} = - 28.0 \ m \ \hat{j}$

Explanation:

Lets say the position of the boat its at the origin of the coordinate system, this is, in vector form:

$\vec{r}_{boat}=0 * \hat{j}$

This simplifies our problem, as the vertical displacement of the diver its:

$\vec{D}=\vec{r}_{diver}-\vec{r}_{boat}$

Now, in this coordinate system, this is the same as:

$\vec{D}=\vec{r}_{diver}$

Be careful. This is only valid in this particular coordinate system.

Ok, we can return to our problem.

Starting at position

$\vec{r}_{diver} = 0 \ \hat{j}$

He first descends 8.0 m, this is :

$\vec{r}_{diver} = 0 \ \hat{j} - 8.0 \ m \ \hat{j}$,

as the unit vector ĵ points upwards.  Making the substaction:

$\vec{r}_{diver} = - 8.0 \ m \ \hat{j}$,

After this, the diver ascends 5.0 m:

$\vec{r}_{diver} = - 8.0 \ m \ \hat{j}  + 5.0 \ m \ \hat{j}$,

$\vec{r}_{diver} = - 3.0 \ m \ \hat{j}$

Then, he continues descending for another 12.0 m:

$\vec{r}_{diver} = - 3.0 \ m \ \hat{j} - 12.0 \ m \ \hat{j}$

$\vec{r}_{diver} = - 15.0 \ m \ \hat{j}$

The, ascends 4.0 m and then descends for 14.0 m, ascends again for 8.0 m and descends again for 11.0 m, all this, will be :

$\vec{r}_{diver} = - 15.0 \ m \ \hat{j} + 4.0 \ m \ \hat{j} - 14.0 \ m \ \hat{j} + 8.0 \ m \ \hat{j} - 11.0 \ m \ \hat{j}$

$\vec{r}_{diver} = - 28.0 \ m \ \hat{j}$

And this is the total vertical displacement.