2-D Motion: Projectiles at an Angle Physics

Transcription

1 -D Motion: Projectiles at an Angle Physics Be sure your calculator is set to DEGREES! I. Trigonometry Reiew: 1. Find the alues of the following functions. (Use scientific calculator) i) sin45º ii) cos40º iii) tan45º i) tan0º ) cos30º i) sin90º ii) cos0º iii) sin0º ix) cos60º. Find the alues of a, b, c or θ c b θ a i) Find a and b when θ=60º and c=10 ii) Find a and b when θ =45º and c=6 iii) Find c and b when θ =50º and a=5 i) Find a and c when θ =0º and b=10 ) Find the θ when c=4 and b= i) Find the θ when c=4 and a=

2 3. Find the x or y components of the following ectors (displacement ector r & elocity ector ) i) Displacement ector r The magnitude of displacement ector r is r =10m at an angle 60º aboe the x-axis. Find the r x, r y components of ector r y (m) r ry r x = ( )m, r y = ( )m θ=60 x(m) ii) Velocity ector An object is thrown at an angle of 40º with the initial speed of 10m/s. What are the initial horizontal speed and the initial ertical speed? y (m) rx x = ( )m/s, y = ( )m/s θ=40 x(m) II. -Dimensional Motion *~ Projectile motion thrown at an angle ~* ymax (when =0) i i θi i (initial horizontal speed) θf (=-θi) initial ertical speed * ~ = cosθ, = sinθ ~* = x + y

3 *~Important features of projectile motion at an angle~* i) The horizontal speed x remains constant throughout the motion => Conceptual analysis : Since there is no horizontal force acting on the object the moment it is release, x remains constant throughout the motion => Mathematical analysis: The horizontal speed at any point can be expressed as x = cosθ, where is actual speed. As the object rises, the actual speed decreases, and at the same time the angle between the actual speed and x decreases. If the angle θ of a cosine function decreases, then the cosθ increases. ii) The ertical speed y decrease as the object rises, and then becomes instantaneously zero at the maximum height. Then y reerse direction and increase in the negatie direction(down) as the object falls towards the surface => Conceptual analysis : The only force acting on the object once release in the air is the force of graity that is pulling the object towards the surface. This causes the object to slow down when rising and speed up when falling => Mathematical analysis: The ertical speed at any point can be expressed as y = sinθ, where is actual speed. As the object rises, the actual speed decreases and sinθ decrease. At the maximum height, since the θ=0, y =0. When the object is falling, the actual speed makes an angle below the x-axis making the θ negatie. This results in y < 0 when falling. *~Formulas for projectile motion at an angle~* - The following formulas are the three general formulas used to describe the projectile f = i + at (1) d= i t + (½)at () ad= f i (3) i) X-component Since there is no acceleration in the horizontal direction, a x=0 and d x. Formula () can rewritten as x = x t where x = cosθ ii) Y-component Since graity pulls the projectile downward, the projectile accelerates downward at a y = -g and d y. The 3 formulas (1), () and (3) can be rewritten as yf = yi + gt, y = yi t + (1/)gt, gy = yf - yi

4 X-component : x = x t where x = cosθ Y-component : yf = yi + gt, y = yi t + (1/)gt, gy = yf - yi *~ Note that the angle θ is not constant. The angle between the initial speed and x-axis is maximum at the beginning of the throw, becomes smaller as the projectile goes up, becomes θ=0 at the maximum height, then increases negatiely as it falls, where θ f =- θ i Finding the maximum height Ex) An object is thrown up at angle of θ=30º with the initial speed of 10m/s. Find the maximum height of the object. Finding the time during the air Ex) An object is shot at angle of θ =30º with the initial speed of 30m/s. How long was the object in the air? Finding the horizontal distance (At what angle does a projectile trael the farthest?) i) An object is thrown up at angle of θ=60º with the initial speed of 10m/s. How far did the object trael? ii) An object is thrown up at angle of θ=45º with the initial speed of 10m/s. How far did the object trael? ii) An object is thrown up at angle of θ =30º with the initial speed of 10m/s. How far did the object trael?

5 X-component : x = x t where x = cosθ Y-component : yf = yi + gt, y = yi t + (1/)gt, gy = yf - yi Additional Questions 1. A placekicker kicks a football at an angle of θ = 40 aboe the horizontal axis. The initial speed of the ball is i =m/s. a) Find the maximum height the ball attains b) Determine the time of flight between kickoff and landing c) Calculate the range of the ball d) What is the speed when t=0.5s, 1s, 1.5s, s and.9s and the angles respectiely? answers => a) ymax = 10.m b) Ttotal=.9s c) Range x = 48.8m d) time 0.5s 1s 1.5s s.9s speed 19.m/s 17.4m/s 16.86m/s 17.71m/s m/s angle

6 X-component : x = x t where x = cosθ Y-component : yf = yi + gt, y = yi t + (1/)gt, gy = yf - yi. A ball shoots upward at an angle of 40 to the horizontal. The initial speed of the ball is 0m/s. (a) How high up will it strike a wall which is 8m away? (b) What is the speed the moment it strikes the wall? (c) Does the ball hit the wall when it is rising or descending? Wall i x answer: (a) y=5.3m (b) =17. m/s (c) rising