Physics 1A. Lecture 3B. "More than anything else... any guy here would love to have a monkey. A pet monkey." -- Dane Cook

Transcription

1 Physics 1A Lecture 3B "More than anything else... any guy here would love to have a monkey. A pet monkey." -- Dane Cook

2 Trajectories Since there is no horizontal acceleration (a x = 0) the horizontal position, x, at any given time can be described by: x = v t = v cos( θ )t ox o o Assuming up is positive, then we can say that the vertical position, y, at any given time is: y = v t + 1 oy 2 a yt 2 = v sin( θ )t 1 o o 2 gt 2 As almost always with projectile motion, we should now solve our x-equation for time to input into the y-equation: t = x v o cos( θ ) o

3 Trajectories Very often we are interested in mathematically describing the trajectory of a projectile. We have already noted that it will follow a parabolic path; so we know that the vertical displacement, Δy, will vary with the square of the horizontal displacement, Δx. Let s attempt to describe the vertical position, y, of a projectile launched with an initial speed v o at an initial angle θ o with the horizontal.

4 Trajectories The y-equation then becomes: y = v o sin θ o ( ) x v o cos θ o y = x tan( θ ) o ( ) 1 2 g x v o cos θ o ( ) We find that the vertical position does indeed depend on the square of the horizontal position. This equation neglects air resistance or any other kind of influence. g 2v o 2 cos 2 θ o ( ) x 2 2

5 Range Often times we would also like to know how far horizontally that a projectile can travel. Starting with our trajectory equation and find out when y = 0 (this assumes you are on level ground): 0 = x tan( θ ) o 0 = x tan( θ ) o We find that one answer is when x = 0, at the beginning of our motion. g 2v 2 o cos 2 ( θ ) x 2 o g 2v 2 cos 2 ( θ ) x o o

6 Range We find that the other answer occurs when argument in the parentheses is zero. 0 = tan( θ ) o tan( θ ) o = x = 2v 2 o g cos2 ( θ o )tan θ o g 2v 2 o cos 2 θ o g ( ) x 2v 2 o cos 2 ( θ ) x o ( ) = 2v 2 o g cos ( θ o)sin θ o ( ) Since sin2θ = 2sinθ cosθ, this becomes: x = v 2 o g sin ( 2θ ) The range o equation.

7 Range The range equation only works for level ground. Without air resistance, the maximum range of a projectile is achieved with an angle of 45 o. Note that the heights of the differing angles will vary dramatically.

8 Range Example A pirate ship is 560m from a fort defending the harbor entrance of an island. A defense cannon at the fort, located at sea level, fires cannonballs at initial speeds of v o = 82m/s. At what angle θ o from the horizontal must a ball be fired to hit the ship? Answer First, you must define a coordinate system. With using the Range Equation, you are assuming that up is +y and the direction of the cannonball s motion as +x.

9 Answer Range Start with the Range Equation: x = v 2 o g sin ( 2θ ) o Since we are looking for the angle, we must use the inverse sine. θ o = 1 2 sin 1 sin( 2θ ) o = xg xg v o 2 xg 2 2θ v o = sin 1 2 o v o = 1 2 sin 1 ( ) 560m ( ) 9.8 m s 2 ( 82 m ) 2 s = 1 2 sin ( ) θ = 1 ( o ) = 27

10 Answer Range But we are not done, as there are two possible values in a unit circle that give us for sine and ( ) = θ o = ( ) = 63 When using the range equation, watch out for the 2θ part. It could give you trouble.

11 Uniform Circular Motion When acceleration is always perpendicular to velocity, then only the direction of velocity changes and not the magnitude. If the acceleration is only perpendicular to the velocity and never colinear then this leads to uniform circular motion. This acceleration is known as radial acceleration, a r, or centripetal acceleration, a c. This acceleration will point towards the center of the circular path of the motion.

12 Centripetal Acceleration Centripetal is another way of saying centerseeking. Even though the magnitude of centripetal acceleration is given by: a c = (v 2 )/r the direction of centripetal acceleration always points toward the center of the circular motion. This radial acceleration component tells us that the body is traveling in a circular type motion (this means that the velocity vector is changing direction).

13 Circular Motion The second type of acceleration is tangential acceleration, a t. This tangential acceleration component tells us if the body is changing its angular speed (this means that v is changing magnitude). a r a t a a t a r

14 Circular Motion The only way to have uniform circular motion is to only have a radial component to acceleration and no tangential component. A radial acceleration only changes direction of the velocity vector, while a tangential acceleration only changes magnitude of the velocity vector.

15 Relative Motion Relative motion is about relating measurements of two different observers. Usually we make measurements with respect to a stationary frame (such as the ground). But sometimes it may be useful to use a moving frame of reference instead of a stationary one. There are no specific equations to learn to solve relative velocity problems. But you will have to use previous equations and be very careful with your notation.

16 Relative Motion Keeping track of your subscripts will be the key to relative motion (although the book doesn t). Let s say that there are two cars (A and B) moving with different speeds in different directions. An observer stationary with respect to the Earth (E) is where we normally take our measurements. But we may need to take measurements of A with respect to B (or vice versa).

17 Relative Motion The position of car A as measured by E is: The position of car B as measured by E is: The position of car A as measured by car B is: We can relate these positions by the following equation: We can then relate velocities by:

18 Relative Motion Example A boat s speed in still water is 20.0km/hr. If the boat is to travel directly across a river whose current has a speed of 12.0km/hr, at what upstream angle must the boat head? Answer First, you must define a coordinate system. Let s say that the boat wants to travel directly north and that the river current flows directly west.

19 Relative Motion Answer According to the equation: v bs = v bw + v ws Let s make a quick diagram of the situation: N v ws W v bs θ v bw The boat will point 36.9 o East of North

20 Relative Motion In the book s notation, they use v as the velocity of an object with respect to some reference frame, S. Then, they suppose that there is some reference frame, S, that moves with a velocity, V, compared to reference frame S. Then, the velocity, v, of the object with respect to reference frame S would be given by: v = v - V

21 Relative Motion When we consider different reference frames, we only consider those that are not accelerating with respect to each other. These are called inertial reference frames. a = a According to the principle of Galilean relativity: The laws of motion are the same in all inertial reference frames. This means that all inertial reference frames have the same physical properties.

22 For Next Time (FNT) Finish Chapter 3 homework Start reading Chapter 4