Physics 221 Chapter 4

Equations in 2-D v = dr / dt r = x i + y j v = dx/dt i + dy/dt j a = dv / dt Note: These are vector equations!

Problem1... velocity vector The position of a particle is given as x = 6 t i y = (8 t - t 2 ) j (a) What is the speed of the particle at t= 6 s? (b) When is the particle moving horizontally?

Solution1... velocity vector (a) v = dx/dt i + dy/dt j v = 6 i + (8 -2 t) j At t= 5 s, v = 6 i - 4 j | v | = 7.2 m/s (b) Horizontal motion occurs when the y-component of the velocity is zero. That happens when t = 4s

Projectile Motion 1.Which projectile reaches the ground first? 2.Which is moving faster just before impact? Two objects are launched horizontally with different speeds

Problem 2... Projectile Motion Given h= 20 m. V H = 7 m/s. Calculate: A. Time to hit the ground B. Impact Speed

Solution 2... Projectile Motion h = ½ a t 2 t = 2 s D = (7 m/s)(2 s) = 14 m V v = 0 + at = (10)(2) = 20 m/s V 2 = (20) 2 + (7) 2 V = 21.2 m/s

Problem 3... Soccer Ball A soccer ball is kicked and it is projected with a speed of 20 m/s at an angle of How far away does it land?

Solution 3... Soccer Ball V v = 20 sin 30 = 10 m/s So air time is 1s (up) and 1s(down) = 2 s D = (V H )(t) D = (20 cos 30)(2) D= 34.6 m

Problem 4... Round and round A red car goes counterclockwise around a track at a constant speed (see figure) A. The car is accelerating and Q shows the direction of the acceleration B. The car is accelerating and P shows the direction of the acceleration C. The car is not accelerating P Q

Solution 4... Round and round A red car goes counterclockwise around a track at a constant speed (see figure) A The car is accelerating and Q shows the direction of the acceleration. Centripetal acceleration changes the direction of motion but not the speed Centripetal acceleration points toward the center and is also called radial acceleration P Q

Problem 5... Equation for a r The correct equation for centripetal acceleration is A. a r = v / r 2 B. a r = v r C. a r = v / r D. a r = v 2 / r Hint: Use dimensional analysis

Solution 5... Equation for a r The correct equation for centripetal acceleration is a r = v 2 / r

Problem 6... Faster and faster! A red car speeds up as it goes around a track. A. P shows the direction of the acceleration B. S shows the direction of the acceleration C. Q shows the direction of the acceleration P Q S

Solution 6... Faster and faster! A red car speeds up as it goes around a track. B. S shows the direction of the acceleration There are two types of acceleration Tangential acceleration (a t ) changes the speed Radial acceleration (a r ) changes the direction a = a r + a t (vector addition!) P Q S

Problem 7... Frame of reference A red car is moving East at 30 m.p.h. and a blue car is moving North at 40 m.p.h. Suppose you are the driver of the red car. What do you think is the velocity of the blue car?

Solution 7... Frame of reference Motion is “relative”. From the red car’s F.O.R. the blue car is traveling north at 40 m.p.h. AND West at 30 m.p.h. ! V B = - 30 i + 40 j = 50 m.p.h. at 127 0