4.1 Displacement and velocity vectors (Page 4/7)

University physics volume 1 Page 4 / 7

An example illustrating the independence of vertical and horizontal motions is given by two baseballs. One baseball is dropped from rest. At the same instant, another is thrown horizontally from the same height and it follows a curved path. A stroboscope captures the positions of the balls at fixed time intervals as they fall ( [link] ).

Two identical balls are illustrated at 5 locations at equal time intervals. The balls start at the same vertical position. Green arrows represent the horizontal velocities and purple arrows represent the vertical velocities at each position. The ball on the right has an initial horizontal velocity whereas the ball on the left has no horizontal velocity. The horizontal motion is constant horizontal velocity at all times for both balls. The vertical motion is constant vertical acceleration. Each ball’s vertical velocity is increasing in magnitude and pointing down. At each instant in time, both balls have identical vertical positions and vertical velocities. — A diagram of the motions of two identical balls: one falls from rest and the other has an initial horizontal velocity. Each subsequent position is an equal time interval. Arrows represent the horizontal and vertical velocities at each position. The ball on the right has an initial horizontal velocity whereas the ball on the left has no horizontal velocity. Despite the difference in horizontal velocities, the vertical velocities and positions are identical for both balls, which shows the vertical and horizontal motions are independent.

It is remarkable that for each flash of the strobe, the vertical positions of the two balls are the same. This similarity implies vertical motion is independent of whether the ball is moving horizontally. (Assuming no air resistance, the vertical motion of a falling object is influenced by gravity only, not by any horizontal forces.) Careful examination of the ball thrown horizontally shows it travels the same horizontal distance between flashes. This is because there are no additional forces on the ball in the horizontal direction after it is thrown. This result means horizontal velocity is constant and is affected neither by vertical motion nor by gravity (which is vertical). Note this case is true for ideal conditions only. In the real world, air resistance affects the speed of the balls in both directions.

The two-dimensional curved path of the horizontally thrown ball is composed of two independent one-dimensional motions (horizontal and vertical). The key to analyzing such motion, called projectile motion , is to resolve it into motions along perpendicular directions. Resolving two-dimensional motion into perpendicular components is possible because the components are independent.

Summary

The position function $\vec{r} (t)$ gives the position as a function of time of a particle moving in two or three dimensions. Graphically, it is a vector from the origin of a chosen coordinate system to the point where the particle is located at a specific time.
The displacement vector $Δ \vec{r}$ gives the shortest distance between any two points on the trajectory of a particle in two or three dimensions.
Instantaneous velocity gives the speed and direction of a particle at a specific time on its trajectory in two or three dimensions, and is a vector in two and three dimensions.
The velocity vector is tangent to the trajectory of the particle.
Displacement $\vec{r} (t)$ can be written as a vector sum of the one-dimensional displacements $\vec{x} (t), \vec{y} (t), \vec{z} (t)$ along the x , y , and z directions.
Velocity $\vec{v} (t)$ can be written as a vector sum of the one-dimensional velocities $v_{x} (t), v_{y} (t), v_{z} (t)$ along the x , y , and z directions.
Motion in any given direction is independent of motion in a perpendicular direction.

Conceptual questions

What form does the trajectory of a particle have if the distance from any point A to point B is equal to the magnitude of the displacement from A to B ?

straight line

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Give an example of a trajectory in two or three dimensions caused by independent perpendicular motions.

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If the instantaneous velocity is zero, what can be said about the slope of the position function?

The slope must be zero because the velocity vector is tangent to the graph of the position function.

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Problems

The coordinates of a particle in a rectangular coordinate system are (1.0, –4.0, 6.0). What is the position vector of the particle?

$\vec{r} = 1.0 \hat{i} - 4.0 \hat{j} + 6.0 \hat{k}$

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The position of a particle changes from ${\vec{r}}_{1} = (2.0 \hat{i} + 3.0 \hat{j}) cm$ to ${\vec{r}}_{2} = (−4.0 \hat{i} + 3.0 \hat{j}) cm .$ What is the particle’s displacement?

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The 18th hole at Pebble Beach Golf Course is a dogleg to the left of length 496.0 m. The fairway off the tee is taken to be the x direction. A golfer hits his tee shot a distance of 300.0 m, corresponding to a displacement $Δ {\vec{r}}_{1} = 300.0 m \hat{i},$ and hits his second shot 189.0 m with a displacement $Δ {\vec{r}}_{2} = 172.0 m \hat{i} + 80.3 m \hat{j} .$ What is the final displacement of the golf ball from the tee?

$Δ {\vec{r}}_{Total} = 472.0 m \hat{i} + 80.3 m \hat{j}$

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A bird flies straight northeast a distance of 95.0 km for 3.0 h. With the x -axis due east and the y -axis due north, what is the displacement in unit vector notation for the bird? What is the average velocity for the trip?

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A cyclist rides 5.0 km due east, then 10.0 km $20 °$ west of north. From this point she rides 8.0 km due west. What is the final displacement from where the cyclist started?

$Sum of displacements = −6.4 km \hat{i} + 9.4 km \hat{j}$

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New York Rangers defenseman Daniel Girardi stands at the goal and passes a hockey puck 20 m and $45 °$ from straight down the ice to left wing Chris Kreider waiting at the blue line. Kreider waits for Girardi to reach the blue line and passes the puck directly across the ice to him 10 m away. What is the final displacement of the puck? See the following figure.

An illustration of the situation described in the problem. The goal and the two ice hockey players are drawn as viewed from above. The goal and Girardi are at the origin of an x y coordinate system. A gray arrow representing 20 meters at 45 degrees from the positive x direction is shown, with Kreider drawn near the tip of the arrow. A blue line, parallel to the x axis, is also drawn at the tip of this arrow. A second gray arrow is shown starting at the Kreider’s location, pointing horizontally to the left, and representing a distance of 10 meters. A dark blue arrow is drawn from the goal at the origin to the tip of the second, 10 meter, gray arrow.

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The position of a particle is $\vec{r} (t) = 4.0 t^{2} \hat{i} - 3.0 \hat{j} + 2.0 t^{3} \hat{k} m .$ (a) What is the velocity of the particle at 0 s and at $1.0$ s? (b) What is the average velocity between 0 s and $1.0$ s?

a. $\vec{v} (t) = 8.0 t \hat{i} + 6.0 t^{2} \hat{k}, \vec{v} (0) = 0, \vec{v} (1.0) = 8.0 \hat{i} + 6.0 \hat{k} m/s$ ,
b. ${\vec{v}}_{avg} = 4.0 \hat{i} + 2.0 \hat{k} m/s$

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Clay Matthews, a linebacker for the Green Bay Packers, can reach a speed of 10.0 m/s. At the start of a play, Matthews runs downfield at $45 °$ with respect to the 50-yard line and covers 8.0 m in 1 s. He then runs straight down the field at $90 °$ with respect to the 50-yard line for 12 m, with an elapsed time of 1.2 s. (a) What is Matthews’ final displacement from the start of the play? (b) What is his average velocity?

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The F-35B Lighting II is a short-takeoff and vertical landing fighter jet. If it does a vertical takeoff to 20.00-m height above the ground and then follows a flight path angled at $30 °$ with respect to the ground for 20.00 km, what is the final displacement?

$Δ {\vec{r}}_{1} = 20.00 m \hat{j}, Δ {\vec{r}}_{2} = (2.000 \times 10^{4} m) (cos 30 ° \hat{i} + sin 30 ° \hat{j})$
$Δ \vec{r} = 1.700 \times 10^{4} m \hat{i} + 1.002 \times 10^{4} m \hat{j}$

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Source: OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5

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