Relative velocity of a swimmer

In summary, the person must swim in a direction of [N 35° W] in order to land directly north of their starting position after crossing a river 84m wide with a current moving at a velocity of 0.40m/s[E]. This is because the swimmer's velocity relative to the water must be equal and opposite to the current's velocity in order to remain stationary relative to the earth.
  • #1
Balsam
226
8

Homework Statement


A person decides to swim across a river 84m wide that has a current moving with a velocity of 0.40m/s[E]. The person swims 0.70m/s[N] relative to the water. In what direction should she swim if she lands at a point directly north of her starting position?

Note: This is part d of the question- in earlier parts we solved for the velocity of the person with respect to the earth: 0.80m/s[N 30° E], time it takes to cross the river: 1.2x10^2s, how far downstream the person will land: 48m.

Homework Equations


v=d/t?

The Attempt at a Solution


I assumed the speed of the person will remain the same: 0.70m/s and the speed and direction of the current will remain constant. I know the person will have to swim in the western direction to so that when the stream pushes them east, they'll land north relative to the starting position. I drew a diagram of this, where the original x and y components are the same(x component is the current's velocity and direction and y component is the swimmer's velocity and direction relative to the water, but the hypotenuse of the right triangle formed by these 2 vectors can be described as having some velocity, with the direction [N θ W], θ being its angle. I tried solving for θ using the tan ratio: tan-1(0.40/0.70)=29.74°. However, the answer in the textbook is [N 35° W]. What did I do wrong?
 
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  • #2
It sounds like you've drawn the swimmers vector triangle wrong. The hypotenuse is the swimmers velocity relative to the water, (in a NW direction), it's x component is equal and opposite to the waters velocity, and the Y component is of no interest to you (swimmers velocity relative to earth).
 
  • #3
billy_joule said:
It sounds like you've drawn the swimmers vector triangle wrong. The hypotenuse is the swimmers velocity relative to the water, (in a NW direction), it's x component is equal and opposite to the waters velocity, and the Y component is of no interest to you (swimmers velocity relative to earth).
Shouldn't the x component be the current? And how do we solve for the angle if we don;t know the y component?
 
  • #4
Balsam said:
Shouldn't the x component be the current?
No, for the swimmer to head due north their net (ie relative to earth)x velocity must be zero, so, relative to the water, the swimmers x velocity needs to cancel the waters x velocity so should be equal and opposite.
Like running on a treadmill, you must run in the opposite direction to the belt to remain stationary relative to earth.

And how do we solve for the angle if we don;t know the y component?
We know two sides (opposite & hypotenuse) so can solve for theta.
 
  • #5
This does work to match the answer in the text but why is the swimmers speed only .7 m/s on the hypotenuse when before they could swim that fast due north? This means they are now only swimming .6 m/s north.
 
  • #6
Iragc said:
This does work to match the answer in the text but why is the swimmers speed only .7 m/s on the hypotenuse when before they could swim that fast due north? This means they are now only swimming .6 m/s north.
The swimmer's speed is always .7m/s relative to the water, no matter in which direction she heads.
It was initially given as .7m/s N relative to the water because in some earlier part of the question that was how she chose to swim. But in that part her velocity relative to the bank was somewhat to the E because of the current.
She now wishes to swim in such a direction relative to the water that her velocity relative to the bank is due N. Since some of her velocity will go into opposing the current, her northerly speed is reduced.
 

Related to Relative velocity of a swimmer

What is relative velocity of a swimmer?

Relative velocity of a swimmer is the speed at which the swimmer moves relative to their surroundings, taking into account both their own speed and the speed of the water they are swimming in.

How is relative velocity of a swimmer calculated?

Relative velocity of a swimmer can be calculated by subtracting their own velocity from the velocity of the water they are swimming in. This can be represented by the equation V(swimmer) = V(water) - V(swimmer).

What factors affect the relative velocity of a swimmer?

The relative velocity of a swimmer can be affected by several factors, including the swimmer's own speed and technique, the current and temperature of the water, and any external forces such as wind or waves.

How does relative velocity differ from absolute velocity?

Relative velocity takes into account the movement of the swimmer relative to their surroundings, while absolute velocity only measures the speed of the swimmer in a straight line. Relative velocity is also affected by external factors, while absolute velocity remains constant.

Why is understanding relative velocity important for swimmers?

Understanding relative velocity is important for swimmers because it allows them to adapt their technique and speed to the conditions of the water they are swimming in. It also helps swimmers to conserve energy and swim more efficiently by using the natural movement of the water to their advantage.

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