Kinematics in Two Dimensions
Addition and subtraction of vectors: geometric method
The vector A shown in Figure
Graphical addition of vectors, A + B = C.
In Figure
Figure
To subtract vectors, place the tails together. The difference of the two vectors ( D) is the vector that begins at the head of the subtracted vector ( B) and goes to the head of the other vector ( A). An alternate method is to add the negative of a vector, which is a vector with the same length but pointing in the opposite direction. The second method is demonstrated in Figure
Graphical subtraction of vectors, A − B = D.
Addition and subtraction of vectors: Component method
For precision in adding vectors, an analytical method using basic trigonometry is required because scale drawings do not give accurate values.
Consider vector A in the rectangular coordinate system of Figure
Components of a vector.
To add vectors numerically, first find the components of all the vectors. The signs of the components are the same as the signs of the cosine and sine in the given quadrant. Then, sum the components in the x direction, and sum the components in the y direction. As shown in Figure
Component method of vector addition, A + B = C.
These resultant components form the two sides of a right angle with a hypotenuse of the magnitude of C; thus, the magnitude of the resultant is
The direction of the resultant ( C) is calculated from the tangent because tan θ = C x / C y . To solve for the angle θ, use θ = tan −1 ( C y / C x ).
The procedure can be summarized as follows:
1. Sketch the vectors on a coordinate system.
2. Find the x and y components of all the vectors, with the appropriate signs.
3. Sum the components in both the x and y directions.
4. Find the magnitude of the resultant vector from the Pythagorean theorem.
5. Find the direction of the resultant vector using the tangent function.
Follow the same procedure to subtract vectors by calculating the appropriate algebraic sum of the components in Step 3.
Multiplication of vectors
The dot product: There are two different ways in which two vectors may be multiplied together. The first is the dot product, also called the scalar product, which is written A · B. This can be evaluated in two ways:
- A · B = A x B x + A y B y
- A · B = AB cos θ, where θ is the angle between the vectors when they are set tail to tail, and A and B are the lengths of the vectors.
Note that the order of the vectors does not matter and that the result of the dot product is a scalar rather than a vector. Note that if two vectors are perpendicular, their dot product is zero according to the second rule above.
Cross product: The second way to multiply vectors is called the cross product or the vector product. It is written A · B. It can be evaluated in two ways:
- A · B = ( A x B y − A y B x z, when the vectors A and B both are in x–y plane. The z indicates that the result is a vector that points along the z axis. In general, the vector resulting from a cross product is always perpendicular to both of the vectors being multiplied together.
- A · B = AB z sin θ, where θ is the angle between the vectors A and B when they are placed tail to tail. Again, the result is a vector perpendicular to A and B (and therefore points along the z axis if A and B are in the x–y plane).
The result of a cross product does depend on the order of the vectors. Note from the first rule that A · B = − B · A. Also, if A and B are parallel, the second rule implies that their cross product is zero.
Finally, the cross product give rise to the “right hand rule,” which allows you to easily determine the direction of the resulting vector. For the general expression A × B = C, point your thumb in the direction of A. Now point your index finger in the direction of B; if necessary, flip over your hand. The vector C points outward from your palm.
Velocity and acceleration vectors in two dimensions
For motion in two dimensions, the earlier kinematics equations must be expressed in vector form. For example, the average velocity vector is v = ( d f − d o )/ t, where d o and d f are the initial and final displacement vectors and t is the time elapsed. As noted earlier, the velocity and displacement vectors are shown in bold type, whereas the scalar (t) is not. In similar fashion, the average acceleration vector is a = ( v f − v o )/ t, where v o and v f are the initial and final velocity vectors.
An important point is that the acceleration can arise from a change in the magnitude of the velocity (speed) as well as from a change in the direction of the velocity. If an object travels around a circle at a constant speed, there is an acceleration due to the change in the direction of the velocity, even though the magnitude of the velocity does not change. A mass moves in a horizontal circle with a constant speed in Figure
- Case 1: Zero acceleration
- Case 2: Acceleration due to changing direction but not speed
- Case 3: Acceleration due to changing speed but not direction
- Case 4: Acceleration due to changing both speed and direction.
Imagine a ball rolling on a horizontal surface that is illuminated by a stroboscopic light. Figure
(a) Path of a ball on a table. (b) Acceleration between points 3 and 4.
Anyone who has observed a tossed object—for example, a baseball in flight—has observed projectile motion. To analyze this common type of motion, three basic assumptions are made: (1) acceleration due to gravity is constant and directed downward, (2) the effect of air resistance is negligible, and (3) the surface of the earth is a stationary plane (that is, the curvature of the earth's surface and the rotation of the earth are negligible).
To analyze the motion, separate the two‐dimensional motion into vertical and horizontal components. Vertically, the object undergoes constant acceleration due to gravity. Horizontally, the object experiences no acceleration and, therefore, maintains a constant velocity. This velocity is illustrated in Figure
With the motions separated into components, the quantities in the x and y directions can be analyzed with the one‐dimensional motion equations subscripted for each direction: for the horizontal direction, v x = v x0 and x = v x0 t; for vertical direction, v y = v y0 − gt and y = v y0 − (1/2) gt 2, where x and y represent distances in the horizontal and vertical directions, respectively, and the acceleration due to gravity ( g) is 9.8 m/s 2. (The negative sign is already incorporated into the equations.) If the object is fired down at an angle, the y component of the initial velocity is negative. The speed of the projectile at any instant can be calculated from the components at that time from the Pythagorean theorem, and the direction can be found from the inverse tangent on the ratios of the components:
Other information is useful in solving projectile problems. Consider the example shown in Figure
Substitution into the horizontal distance equation yields R = ( v o cos θ) T. Substitute T in the range equation and use the trigonometry identity sin 2θ = 2 sin θ cos θ to obtain an expression for the range in terms of the initial speed and angle of motion, R = ( v o 2/ g) sin 2θ. As indicated by this expression, the maximum range occurs when θ = 45 degrees because, at this value of θ, sin 2θ has its maximum value of 1. Figure
Range of projectiles launched at different angles.
Uniform circular motion
For uniform motion of an object in a horizontal circle of radius (R), the constant speed is given by v = 2π R/ T, which is the distance of one revolution divided by the time for one revolution. The time for one revolution (T) is defined as period. During one rotation, the head of the velocity vector traces a circle of circumference 2π v in one period; thus, the magnitude of the acceleration is a = 2π v/ T. Combine these two equations to obtain two additional relationships in other variables: a = v 2/ R and a = (4π 2/ T 2) R.
The displacement vector is directed out from the center of the circle of motion. The velocity vector is tangent to the path. The acceleration vector directed to the center of the circle is called centripetal acceleration. Figure
Uniform circular motion.
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