Lesson Using vectors to solve problems in Mechanics: Velocity

Using vectors to solve problems in Mechanics: Velocity

Problem 1

A rowboat moves across the river.  A rower in the boat heads a direction                                         perpendicular to the river's banks  (Figure 1a).  The boat's speed in this direction is 2 m/s.  The river current speed is 1 m/s. Draw the vector of the full velocity of the boat relative to the river's bank. You can assume that the river's banks are parallel straight lines. Solution The full vector of the boat velocity relative the river's banks has two components. From one side,  the boat moves across the river with oars having the velocity component  2  m/s  perpendicular to the river's banks due to rowing. From the other side,  the boat moves along the river having the velocity component 1  m/s  in this direction relative to the river banks due to current.

Figure 1a.  The river, the current and the rowboat crossing the river

Figure 1b.  The full vector of the boat velocity relative to the river banks

Problem 2

A river whose width is  200 m,  has the current speed of  1 m/s. A rowboat is crossing the river.  A rower heads due a direction perpendicular                                  to the river's banks rowing at the speed of  2 m/s  relative to the water. At what distance the rowboat will get the other bank counting downstream from the point  O  opposed to the starting point  D  (Figure 2a)? You can assume that the river's banks are parallel straight lines. Solution The rowboat has the speed of  2 m/s  in the direction perpendicular to the banks due to rowing.  Hence,  it will take   = 100 seconds  to cross the river and to get the opposite bank.

Figure 2a.  The river, the current and the rowboat crossing the river

Figure 2b.  The full vector of the boat velocity relative to the river banks

Problem 3

A man is going to cross the river in a rowboat.  The river's width is  200 m,                                     the current speed is of  1 m/s. A man can row at the speed of  1 m/s  relative to the water. At what direction should the rower head the rowboat to get the other bank at the point A  directly opposed to the starting point  D  (Figure 3a)? You can assume that the river's banks are parallel straight lines. Solution Because the boat is moving partly with the current,  the boat headed due the perpendicular to the river's banks will drift with the current.  To compensate for this cross-flow,  the boat must be headed due some angle against the current as it is shown in the Figures  3a  and  3b.

Figure 3a.  The rowboat crosses               the river overcoming the current

Figure 3b.  The vectors of the                     velocities

Problem 4

In rainy weather the drivers who drive their cars on the highway,  can see the slanting                 raindrops behind the car's side windows  (Figure 4a). Calculate the angle that the raindrops makes with the vertical  (Figure 4b)  as viewed by the driver,  if the raindrop speed near the earth surface is  10 m/s   (meters per second)  and the car's speed is  65  mph  (miles per hour). Assume that raindrops fall vertically down  in the coordinate system connected to the earth. Solution First,  let us convert the speed of  65  mph to meters per second. 1 mile = 1609 m (approximately),  so  65 mph = 1609*65/3600 = 29.05 m/s.

Figure 4a.  The slanting raindrops      behind the car's side window

Figure 4b.  The angle between      the raindrops and the vertical