Physics for Dentistry and Medicine students Physics for Dentistry and Medicine students PHYS 145 Text book Physics; John D. Cutnell and Kenneth W. Johnson; 7 th edition; Wiley; 2007.

Chapter 3 Kinematics in Two Dimensions Once this mountain lion is airborne, it follows a path that depends on the launch velocity and the acceleration due to gravity, assuming that the effects of air resistance can be ignored. (George H. H. Huey/Corbis Images)

Displacement, Velocity, and Acceleration Definition:The displacement is the vector that points from an object’s initial position to its final position and has a magnitude that equals the shortest distance between the two positions. SI Unit of displacement: meter (m)

Average Velocity Definition of Average Velocity SI Unit of Average Velocity and Speed: meter per second (m/s)

Instantaneous Velocity The instantaneous velocity v of the car indicates how fast the car moves and the direction of the motion at each instant of time. The magnitude of the instantaneous velocity is called the instantaneous velocity, and it is the number (with units) indicated by the speedometer

Acceleration Definition of Average Acceleration SI Unit of Average Acceleration: (m/s 2 ) The instantaneous acceleration

Equations of Kinematics in two Dimensions x ComponentVariabley Component x Displacement y axax Acceleration ayay vxvx Final velocity vyvy v ox Initial velocity v oy t Elapsed time t v x = v ox + a x tv y = v oy + a y t x = ½ (v ox + v x ) ty = ½ (v oy +v y ) t x = v ox t + ½ a x t 2 y = v oy t + ½ a y t 2 v x 2 = v ox 2 + 2a x xv y 2 = v oy 2 + 2a y y

Projectile Motion

Important notes: 1. The x component of the velocity remains constant at its initial value or v x = v ox. 2. The x component of the acceleration is a x = 0 m/s The a y is the downward acceleration due to gravity a y = g = 9.80 m/s 2 or 32.2 ft/s The time to reach maximum height from any point = the time spent returning from the maximum height to that point. 5. v y = 0 m/s at maximum height.

Example An airplane moving horizontally with a constant velocity of +115 m/s at an altitude of 1050 m. The directions to the right and upward have been chosen as the positive directions. The plane releases a “care package” that falls to the ground along a curved trajectory. Ignoring air resistance, determine the time required for the package to hit the ground.

Solution y - Direction Data yayay vyvy v oy T m m/s 2 0 m/s? Substitute in the following equation of motion, the falling time can be calculated. y = v oy t + ½ a y t 2 where t = 14.6 s

Example 6: The Height of a Kickoff A placekicker kicks a football at an angle of θ = 40.0 o above the horizontal axis, as in the figure. The initial speed of the ball is v o = 22 m/s. Ignore air resistance and find the maximum height H that the ball attains.

Solution To calculate the vertical component v oy = v o sin θ = + (22 m/s) sin 40.0 o = + 14 m/s v y 2 = v oy 2 + 2a y y y = H = + 10 m y - Direction Data yayay vyvy v oy t H = ? m/s 2 0 m/s+14 m/s

Solution To determine the time of flight between kickoff and landing y = v oy t + ½ a y t 2 where t = 0 s or 2.9 s y - Direction Data yayay vyvy v oy t 0 m m/s m/s?

Solve This Problem 1/307) A jetliner is moving at a speed of 245 m/s. The vertical component of the plane’s velocity is 40.6 m/s. Determine the magnitude of the horizontal component of the plane’s velocity.

H.W. 3/307) A mountain-climbing expedition establishes two intermediate camps, labeled A and B in the drawing, above the base camp. What is the magnitude  r of the displacement between camp A and camp B?

Problem 12/309) On a spacecraft, two engines are turned on for 684 s at a moment when the velocity of the craft has x and y components of v ox = 4370 m/s and v oy = 6280 m/s. While the engines are firing, the craft undergoes a displacement that has components of x = 4.11 x 10 6 m and y = 6.07 x 10 6 m. Find the x and y components of the craft’s acceleration.

Problem 14/309) A volleyball is spiked so that it has an initial velocity of 15 m/s directed downward at an angle of 55° below the horizontal. What is the horizontal component of the ball’s velocity when the opposing player fields the ball?

Quiz Two trees have perfectly straight trunks and are both growing perpendicular to the flat horizontal ground beneath them. The sides of the trunks that face each other are separated by 1.3 m. A frisky squirrel makes three jumps in rapid succession. First, he leaps from the foot of one tree to a spot that is 1.0 m above the ground on the other tree. Then, he jumps back to the first tree, landing on it at a spot that is 1.7 m above the ground. Finally, he leaps back to the other tree, now landing at a spot that is 2.5 m above the ground. What is the magnitude of the squirrel’s displacement?