PROBLEMS
1. When a comet is far from other objects, what can we say about its velocity? Its position? Its acceleration?
2. The position of a coffee cup on a table as referenced by the corner of the room in which it sits is . How far is the cup from the corner? What is the unit vector pointing from the corner to the cup?
3. Make a time-dependent position vector that describes the cup from the previous problem such that it starts in the location given, but slides at a constant rate of 1 m/s in the direction. Show that your position vector gives the right velocity by doing the math.
4. The velocity of a plane is given by . What is the speed of the plane? What is its direction of travel? If the x-direction is east and the y-direction is north, at what angle north of east is the plane flying?
5. Give estimates of how fast you can run, how fast a car drives on the freeway and how fast jet planes fly in meters per second.
6. Given a position function of , what is the velocity function? What is the velocity vector at t=3.0s?
7. What one-dimensional position function must an object have such that the object's position is numerically equal to its speed at all moments in time? In other words if it's at then it's also traveling at that moment. Solve by writing out an appropriate equation first.
8. Given a velocity of what must the units of the numbers '5' and '3' be? What will the displacement of the object described by this function be in the interval between t=1s and t=4s? If the object started at when t=1, where would it be at t=4s?
9. What is the average velocity of a race car that does exactly 440 laps around a one mile circuit in 4.0 hours? What is the magnitude of the average velocity? What is the average speed of the race car?
10. A cyclist races in a straight line with a speed function in meters per second given by , where a,b and c are constants. Define a function in GeoGebra of this form and adjust the constants so that after a long time the cyclist is going 13 m/s, and early on reaches a maximum speed of 20 m/s, 8 seconds from the start. What values of a,b and c accomplish this (with correct units)? How far will the cyclist have gone in 60s? What will the cyclist's speed be at t=10s? How long will it take to cover one kilometer? Please use GeoGebra to do this. At what time is the acceleration maximum? Minimum?
11. An ant runs along a piece of paper following a plot of the function . One unit on the paper is a centimeter. If the ant starts at the origin, how far will it have run when it gets to x=5.0 cm? If it takes 4.0s, what was its average speed? Average velocity? The magnitude of the average velocity? Use GeoGebra to do this problem.
12. Make a plot of the acceleration of a ball that is thrown upward at 20 m/s subject to gravitation alone (no drag). Assume upward is the +y direction (and downward negative y).
13. Make a plot of the same ball's velocity.
14. Make a plot of the same ball's position.
15. At what time will the ball be back at its starting position?
16. At what time will the same ball be at max height?
17. What will the velocity in the y direction be at max height?
18. Do any of these answers change if the ball additionally has a horizontal component to its initial velocity?
19. Name these variables:
20. A car driving at 27m/s veers to the left to avoid a deer in the road. The maneuver takes 2.0s and the direction of travel is altered by 20 degrees. (Don't worry about the fact that they will later veer back into their own lane.) What is the average acceleration during the constant speed maneuver? Do this in accordance with the example in the chapter.
ANSWERS
1. Constant velocity, potentially changing position, zero acceleration.
2. 5.59m,
3.
4. 206.1 m/s; ; 14 degrees.
5. 8m/s; 30m/s; 300m/s
6. ;
7. with A being any constant.
8. m/s; m/s/s; -7.5m;
9. (zero vector), zero, 110 mph
10. a=0.81, b=0.25, c=13, 884m, 19.65m/s, 68.95s, 2.34s, 13.66s
11. 25.87cm, 6.47cm/s, , 6.37cm/s.
12. Acceleration plot is constant at -10m/s2.
13. Linearly decreasing plot. y-intercept 20m/s, x-intercept 2.0s.
14. Downward facing parabola.
15. t=4.0s
16. t=2.0s
17. 0m/s
18. see text
19. see text
20. to the left. (direction must be specified since acceleration is a vector)