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Topic 1.3 Extended B - Components of motion Motion in Two Dimensions 3-1 Components of motion y y FYI: The displacement vector gives Up to now we have considered objects moving in one the direction of the motion dimension. However, most objects move in more than one We dimension. can sketch in our x and y for successive snapshots to obtain an idea ofhere: For example, consider the ball shown the different velocities the ball has at different times: x is in YELLOW. y is in RED. We can also sketch in the displacement d of the ball at each time interval (in GREEN). Let's examine one time interval in detail: x Topic 1.3 Extended B - Components of motion x displacement triangle vy If we know the time interval t between snapshots, we can find the velocity of the ball simply by dividing the displacements shown above by t. The proportions of our triangle will not change. Thus Magnitude of a v = vx2 + vy2 2D velocity y Each triangle gets a good name: vy we can find the value of d if we know x and y: d2 = x2 + y2 Magnitude of a d = x2 + y2 2D displacement y From the Pythagorean Theorem vx velocity triangle sin θ = opposite θ adjacent opp vy hyp v cos θ = vy component of the velocity. We call the vy the vertical component of the velocity. From trigonometry we know there is a relationship between the sides of a triangle, and the angle : vy We call the vx the horizontal vx horizontal component adj vx opp vy tan θ = hyp v adj vx vy = v sin θ vx = v cos θ s-o-h-c-a-h-t-o-a trigonometric ratios vertical component Topic 1.3 Extended B - Components of motion vx = (25.0 m/s)cos 30° vy of the ball is 25.0 m/s at an angle of 30° with respect to (wrt) the positive x-axis. What is vx the horizontal component of the velocity? vx = v cos θ vy Suppose we know the velocity vx vx = v cos θ vx = 21.7 m/s What is vy the vertical component of the velocity? vy = v sin θ vy = (25.0 m/s)sin 30° vy = 12.5 m/s FYI: You can check your results by squaring each answer, summing, and taking the square root. What should you get? vy = v sin θ Topic 1.3 Extended B - Components of motion Topic 1.3 Extended B - Components of motion vy components of the velocity, and want to find the magnitude and the direction: Suppose vx = 30.0 m/s. Suppose vy = 40.0 m/s. Then v = vx2 + vy2 v = 302 + 402 v = 50.0 m/s and opp tan θ = adj vy = vx θ = 4 3 vx magnitude of v = 40 m/s 30 m/s so that tan-1 vy Sometimes we know the = 53.1° direction of v Topic 1.3 Extended B - Components of motion Sometimes we know the formulas for the components of the velocity of a ball, and want to find the magnitude and the direction of the velocity at a particular time: Suppose vx = 30.0 (measured in m/s). Suppose vy = 40.0 - 5t (vy in m/s, t in s) Then what is the velocity at t = 2 s? vx = 30.0 m/s v = vx2 + vy2 vy = 40 - 5(2) v = 302 + 302 vy = 30.0 m/s v = 42.4 m/s magnitude of v What is the direction of the ball at this instant? vy 30 m/s opp tan θ = so that adj = vx = 30 m/s θ = tan-1(1) = 45.0° direction of v