Topic 1.3 Extended B - Components of motion

Up to now we have considered objects moving in one dimension. However, most objects move in more than one dimension. For example, consider the ball shown here: Motion in Two Dimensions 3-1 Components of motion We can sketch in our x and y for successive snapshots to obtain an idea of 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 y d y FYI: The displacement vector gives the direction of the motion

From the Pythagorean Theorem we can find the value of d if we know x and y: d 2 = x 2 + y 2 Topic 1.3 Extended B - Components of motion x y d y d = x 2 + y 2 Magnitude of a 2D displacement 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. vxvx vyvy v vyvy Thus v = v x 2 + v y 2 Magnitude of a 2D velocity Each triangle gets a good name: displacement triangle velocity triangle

We call the v x the horizontal component of the velocity. Topic 1.3 Extended B - Components of motion vxvx vyvy v vyvy horizontal component We call the v y the vertical component of the velocity. vertical component From trigonometry we know there is a relationship between the sides of a triangle, and the angle : opp hyp adj hyp opp adj hypotenuse adjacent opposite θ trigonometric ratios s-o-h-c-a-h-t-o-a v v x = v cos θ v y = v sin θ v vxvx vyvy v sin θ = cos θ = tan θ = vxvx vyvy

Suppose we know the velocity of the ball is 25.0 m/s at an angle of 30° with respect to (wrt) the positive x-axis. Topic 1.3 Extended B - Components of motion vxvx vyvy v vyvy What is v x the horizontal component of the velocity? v x = v cos θ v y = v sin θ v x = v cos θ v x = (25.0 m/s)cos 30° v x = 21.7 m/s What is v y the vertical component of the velocity? v y = v sin θ v y = (25.0 m/s)sin 30° v y = 12.5 m/s FYI: You can check your results by squaring each answer, summing, and taking the square root. What should you get?

Sometimes we know the components of the velocity, and want to find the magnitude and the direction: Topic 1.3 Extended B - Components of motion vxvx vyvy v vyvy Suppose v x = 30.0 m/s. Suppose v y = 40.0 m/s. Then v = v x 2 + v y 2 v = v = 50.0 m/s magnitude of v opp adj tan θ = vxvx vyvy = 40 m/s = 30 m/s and so that θ = tan = 53.1° direction of v

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: Topic 1.3 Extended B - Components of motion Suppose v x = 30.0 (measured in m/s). Suppose v y = t (v y in m/s, t in s) Then what is the velocity at t = 2 s? v = v x 2 + v y 2 v = v = 42.4 m/s magnitude of v opp adj tan θ = vxvx vyvy = 30 m/s = What is the direction of the ball at this instant? so that θ = tan -1 (1) = 45.0° direction of v v x = 30.0 m/s v y = (2) v y = 30.0 m/s