MATH AND MEASUREMENT
1. Given , calculate x.
2. Given = 10, calculate v.
[0.995c]
3. Given = 8, calculate b.
4. Given x-0.2 = 2, calculate s.
5. Given = 15, calculate x.
6. Given 4-4/x = -1/x2, calculate x.
7. Light travels at 3´108m/s. Express its speed in units of mm/nsec.
[3´105]
8. How many nsec are there in a century (100yr)?
9. How many kg are there in a ng?
10. A fortnight is 14 days, a furlong is 220 yards. 22 m/s is how many furlongs/fortnight?
11. If light travels 3´108 m/s express its speed in units of mm/nsec.
12. How many mm are there in an inch?
13. How many nsec are there in a year?
14. Express ‘g’ in units of Mm/(ksec)2.
15. Dave Barry has written about the Hubble Space Telescope that “it would have been cheaper to take a regular telescope and put it on top of an 87-mile-high pile of $50 bills.” Estimate how much money would be in such a pile. (This is an order-of-magnitude estimate: be aware of how many digits are significant.)
ONE-DIMENSIONAL KINEMATICS
1 A ball travels in a vertical direction and lands with a speed of 6m/s.
a) What might the initial velocity have been, if the ball was thrown from a height of 1m?
b) How long would it have taken the ball to fall to the ground if it had started from rest and ended with a speed of 6m/s?
2. One car travelling on a test track, accelerating at –3m/s2, can stop in a distance of 400m. A second car travelling at 50m/s tries to stop in 400m, but still has a speed of 5m/s at the end of 400m.
a) What is the initial velocity of the first car?
b) What is the acceleration of the second car?
[49m/s, -3.1m/s2]
3. To avoid concussion in a a car accident, you need to accelerate less than about l0g or 98m/s2. To keep from being crashed, you need the car to travel less than the length of the hood, about 1.5m.
a) How fast can a car be travelling before crashing into a brick wall, in order to avoid exceeding either limit?
b) How long does such a collision last?
4. You throw a ball straight up in the air from ground level in such a manner that it takes 2s for it to reach its maximum height.
a) What was its initial velocity?
b) How high did it travel?
c) How long does it take until it hits the ground again?
[20m/s, 4s]
5. Carl Lewis, the Olympic sprinter, was timed as taking 1.88s to sprint 10m, starting from rest. Assuming a constant acceleration,
a) What was his average acceleration?
b) What was his final speed?
6. During a car crash, a car starting at 30mph slows to 0mph in a distance of 0.6m.
a) What is the average acceleration of the car in m/s2? (One mile is 1609m.)
b) How many seconds does the collision last?
[150m/s2, 0.089s]
7. A driver begins at rest at t=0s. She accelerates uniformly to 25m/s in 5s, then decelerates to 0m/s in the next 10s, reverses gear and accelerates uniformly to a backwards 25m/s in 7s, travels 3s at this spced and then brakes uniformly to 0m/s in 5s.
a) What is her average acceleration over the first 22s?
b)What is her instantaneous acceleration at 11s?
c) What is her net displacement after all this driving?
[-1.1m/s2, -2.5m/s2, -37.5m]
8. The area under a velocity vs time curve is equal to ...
a) maximum velocity
b) maximum acceleration
c) minimum acceleration
d) displacement
e) It has no physical significance.
9. A car goes from 0 to 80 km/hr in 6 seconds.
a) What is its acceleration (in MKS units)?
b) How are does it travel in 6 seconds?
c) How far would it take for the car to go from 0 to 80 km/hr in free fall?
10. A car travels with a speed v. With its wheels locked and skidding, its minimum stopping distance is d. Assuming that all other factors remain constant, tripling the initial speed will increase this stopping distance to ...
11. A ball rolls down an inclined plane with constant acceleration, starting from rest. Distances are marked every 3s, and the second mark after the starting point is 2.3m from the starting point.
a) What is the acceleration?
b) Where is the first mark?
c) Where is the third mark?
d) What is its speed after it travels 2m?
12. Fill in the following blanks with a verbal description of what each of the items in Fig. E represents:
a) The slope of line A represents ______
b) The slope of line B represents ______
c) The slope of line C represents ______
d) The shaded area D represents ______
Note: by ‘slope’ I mean a number that is positive or negative. I do not mean the magnitude of the slope.
TWO-DIMENSIONAL KINEMATICS
Vectors
1. Add these two vectors analytically. Find the magnitude and direction of the sum.
Vector A: magnitude =10m/s, at 45°
Bx=2m/s; By =5m/s
2. Subtract these two vectors analytically. Find the magnitude and direction of the difference.
Px=3m/s2; Py= -2m/s2
Vector Q: magnitude = 5m/s2 at 1800
[8.2m/s2 at -14o]
3. A hiker travels 13 mi in a direction 30° east of due north, then 5 mi in a direction 60° north of due west.
a) What is the hiker’s net displacement?
b) Give directions to the hiker to get to a point 20 mi due north of the starting point. (For example, “Go 20 miles North, then 5 miles East.”)
4. Thelma and Louise drive their car off a 75m high cliff. The car starts with a horizontal velocity of 80 km/hr.
a) How many seconds later does the car hit the ground?
b) What is the diagonal distance between the edge of the cliff and the impact site?
c) What is the car’s final speed in m/s?
[3.91s, 115m, 44m/s]
5. A polar bear, seeking a yummy harp seal meal, travels first 50 miles due East, then 75 miles at 230°. How far and in what direction must he travel in order to return to his starting point7
[57mi at 92o]
6. A runner travels for 60s at 4 m/s due north, for 100s at 3.5 m/s at 30° north of due east, and for 120s at 4 m/s due west. Calculate
a) the runner’s net displacement,
b) the runner’s average velocity,
c) the runner’s average speed,
d) the runner’s average (nonzero) acceleration.
7. A hiker travels 13 mi in a direction 30° east of due north, then 5 mi in a direction 60° north of due west.
a) What is the hiker’s net displacement?
b) Give directions to the hiker to get to a point 20 mi due north of the starting point.
8. If I begin a journey 11 mi north of Potsdam and end, 30 minutes later, 15 mi at 15° south of west of Potsdam, what is the magnitude and direction of my average velocity?
9. When a vector is resolved into components, they must be
a) perpendicular to each other
b) in the vertical and horizontal directions
c) of unequal lengths
d) All of the above must be true
e) None of the above is necessarily true.
10. Given the two vectors, , with a magnitude of 2m and a direction of 225° from the positive x-axis, and , with a magnitude of 3m and a direction of -30°, sketch the following (use a separate sketch for each) and solve for magnitude and direction of:
Two-dimensional kinematics
11. A car travels due East from Canton to Potsdam (We'll call this the positive x-direction.), covering 11 miles in 15 minutes. Once there, it travels 15 miles due North (toward Massena) in 18 minutes. Assuming both parts of the trip to be constant velocity, and assuming that the car is in motion at the start and end of the trip,
a) What is the magnitude of the average velocity (in m/s) of the entire trip?
b) What is the magnitude of the average acceleration (in m/s2)?
12. A runner travels for 60s at 4 m/s due north, for 100s at 3.5 m/s due east, and for 120s at 4 m/s due south. Calculate
a) the runner’s net displacement,
b) the runner’s average velocity,
c) the runner’s average speed,
d) the runner’s average (nonzero) acceleration.
[420m at -34o, 1.5m/s at -34o, 3.8m/s, -0.029m/s2 due S]
13. A runner travels around a circular track of radius 64m at a uniform speed of 6.7m/s. It takes her 30s to travel halfway around the track. If her starting point is at the Easternmost point on this circle, and if she maintains a constant speed throughout, and we use the Eastern direction as the positive x-axis, and North is the positive y-direction, then calculate -- for the first 30s of her journey --
a) the magnitude and direction of her average velocity.
b) the magnitude and direction of her average acceleration.
[4.3m/s West, 0.44m/s2 South]
14. A cannon is fired horizontally from the edge of a cliff, 150m above sea level. The cannonball reaches the ocean 150m away from the base of the cliff. What is the initial velocity of the cannonball?
15. You pitch a ball at 28m/s, 45° above the horizon from a second story window 5m above the ground.
a) What are the components of the intial velocity?
b) How long does it take for the ball to hit the ground?
c) How far from the base of the building does the ball land?
16. A tennis player hits a ball 100 below the horizon at 30m/s, from a height of 2.5m.
a) How long does it take the ball to hit the ground?
b) What are the components of the velocity the instant before the ball hits the ground?
[0.36s, 30m/s and -9m/s]
17. A tennis player serves a tennis ball so that its velocity as it leaves the racket is totally horizontal. The ball is 2.5m above the ground when served. The ball must land within about 15m from the server for the serve to be inbounds.
a) How fast may the tennis player hit the ball (maximum speed)?
b) How long does it take the ball to reach the ground?
18. A diver jumps from a diving platform into a pool.
a) If the diver's initial velocity was 4m/s at an angle of 30° above the horizontal, what are the
components of the intial velocity?
Assume now that the platform was 4m above the level of the pool and that the diver jumps at 4m/s in a horizontal direction.
b) How many seconds does the dive last?
c) How far from the base of the platform does the diver land?
[3.5m/s, 2m/s, 0.90s, 3.6m]
19. Adiver dives from a platform 10m above water. She leaves the diving board with a velocity such that vox=2m/s and voy= 10m/s.
a) What are the components of the diver's final velocity as she enters the water?
b) How long is she in the air?
c) How far has she traveled in the horizontal direction when she enters the water?
[2m/s, -17m/s, 2.8s, 5.6m]
20. For the ‘monkey-and-the-blowdart’ demonstration, a gun is set at 1.5m above the floor. A ball is fired that must travel 8m horizontally to hit a can. The ball is fired with an initial velocity vo at a 45° angle. Calculate the minimum value for vo such that the ball hits the can before it hits the ground.
21. A cannon is fired horizontally from the edge of a cliff, 150m above sea level. The cannonball reaches the ocean 150m away from the base of the cliff. What is the initial velocity of the cannonball?
22. Show that, for a projectile fired from the ground with an initial speed, vo, fired at an angle q, the range is given by
23. Thelma and Louise drive their car off a 75m high cliff. The car starts with a horizontal speed of 80 km/hr.
a) How many seconds later does the car hit the ground?
b) At what angle does the car hit the ground?
c) What is the car’s final speed in km/hr?
FORCES AND FREE-BODY DIAGRAMS
1. Acceleration is produced by
a) inertia
b) velocity
c) mass
d) force
e) any of the above
2. It was once thought that space travel would be achieved by firing astronauts out of a gigantic cannon. (They would, of course, be inside some sort of space vessel when fired.) To achieve the necessary 11km/s escape speed, what average force would each 70kg astronaut need to experience inside a hypothetical 2.2km long gun?
[1.9MN or 200tons]
3. A 600kg sailboat experiences two forces, a northerly one from the wind and an easterly one due to the water. The net acceleration is 0.5m/s2 in a direction 60o North of East.
a) What force is exerted by the wind? by the water?
b) If the boat starts with a speed of 3m/s due West, what is its displacement vector after three seconds of uniform acceleration?
[150N, 260N, 8m at 104o]
4. Two 5kg bags of sugar are hanging from the ceiling of an elevator car. A massless rope attached to the ceiling holds the top bag, and a massless chain hanging from the top bag supports the lower bag. What are the tensions in the rope (T1) and the chain (T2) if the elevator is travelling upward but slowing down at a rate of 2m/s per second?