Kinematics

A swan on a lake gets airborne by flapping its wings and running on top of the water. 

(a) If the swan must reach a velocity of 6.00 m/s to take off and it accelerates from rest at an average rate of 0.350 m/s², how far will it travel before becoming airborne? 

(b) How long does this take?

A woodpecker’s brain is specially protected from large decelerations by tendon-like attachments inside the skull. While pecking on a tree, the woodpecker’s head comes to a stop from an initial velocity of 0.600 m/s in a distance of only 2.99 mm. 

(a) Find the acceleration in m/s² and in multiples of (g = 9.80 m/s²).

(b) Calculate the stopping time. 

(c) The tendons cradling the brain stretch, making its stopping distance 4.50(greater than the head and, hence, less deceleration of the brain). What is the brain’s deceleration, expressed in multiples of g?

An unwary football player collides with a padded goalpost while running at a velocity of 7.50 m/s and comes to a full stop after compressing the padding and his body 0.350. 

(a) What is his deceleration? 

(b) How long does the collision last? 

Consider a grey squirrel falling out of a tree to the ground. 

(a) If we ignore air resistance in this case (only for the sake of this problem), determine a squirrel’s velocity just before hitting the ground, assuming it fell from a height of 3.0 m. 

(b) If the squirrel stops in a distance of 2.0 cm through bending its limbs, compare its deceleration with that of the airman in the previous problem.

Dragsters can actually reach a top speed of  in only  considerably less time than given in Example 2.10 and Example 2.11. 

(a) Calculate the average acceleration for such a dragster. 

(b) Find the final velocity of this dragster starting from rest and accelerating at the rate found in (a) for 402 m (a quarter mile) without using any information on time. 

(c) Why is the final velocity greater than that used to find the average acceleration? 

Hint: Consider whether the assumption of constant acceleration is valid for a dragster. If not, discuss whether the acceleration would be greater at the beginning or end of the run and what effect that would have on the final velocity.

A bicycle racer sprints at the end of a race to clinch a victory. The racer has an initial velocity of 11.5 m/s and accelerates at the rate of 0.500 for 7.00 s. 

(a) What is his final velocity? 

(b) The racer continues at this velocity to the finish line. If he was 300 m from the finish line when he started to accelerate, how much time did he save? 

(c) One other racer was 5.00 m ahead when the winner started to accelerate, but he was unable to accelerate, and travelled at 11.8 m/s until the finish line. How far ahead of him (in meters and in seconds) did the winner finish?

A swimmer bounces straight up from a diving board and falls feet first into a pool. She starts with a velocity of 4.00 m/s, and her take-off point is   above the pool. (a) How long are her feet in the air? (b) What is her highest point above the board? (c) What is her velocity when her feet hit the water?

In 1967, New Zealander Burt Munro set the world record for an Indian motorcycle, on the Bonneville Salt Flats in Utah, with a maximum speed of\({\bf{183}}.{\bf{58}}{\rm{ }}{\bf{mi}}/{\bf{h}}\). The one-way course was \({\bf{5}}.{\bf{00}}{\rm{ }}{\bf{mi}}\) long. Acceleration rates are often described by the time it takes to reach \({\bf{60}}.{\bf{0}}{\rm{ }}{\bf{mi}}/{\bf{h}}\) from rest. If this time was \({\bf{4}}.{\bf{00}}{\rm{ }}{\bf{s}}\), and Burt accelerated at this rate until he reached his maximum speed, how long did it take Burt to complete the course?

a) Calculate the height of a cliff if it takes \({\bf{2}}.{\bf{35}}{\rm{ }}{\bf{s}}.\) for a rock to hit the ground when it is thrown straight up from the cliff with an initial velocity of\({\bf{8}}.{\bf{00}}{\rm{ }}{\bf{m}}/{\bf{s}}\). (b) How long would it take to reach the ground if it is thrown straight down with the same speed?

A very strong, but inept, shot putter puts the shot straight up vertically with an initial velocity of \(11.0 m/s\)How long does he have to get out of the way if the shot was released at a height of\(2.20 m\), and he is \({\bf{1}}.{\bf{80}}{\rm{ }}{\bf{m}}\) tall?

A graph of v(t) is shown for a world-class track sprinter in a \({\bf{100}} - {\bf{m}}\) race. (See Figure 2.67). (a) What is his average velocity for the first 4 s? (b) What is his instantaneous velocity at \(t = {\rm{ }}{\bf{5}}{\rm{ }}{\bf{s}}\)? (c) What is his average acceleration between 0 and 4 s? (d) What is his time for the race?

Give an example in which there are clear distinctions among distance traveled, displacement, and magnitude of displacement. Specifically, identify each quantity in your example

Under what circumstances does distance travelled equal magnitude of displacement? What is the only case in which magnitude of displacement and displacement are exactly the same?

Bacteria move back and forth by using their flagella (structures that look like little tails). Speeds of up to 50 µm/s (50×10^-6 m/s) have been observed. The total distance travelled by a bacterium is large for its size, while its displacement is small. Why is this?

A student writes, “A bird that is diving for prey has a speed of $\mathbf{-}\mathbf{10}\mathbf{\text{\hspace{0.17em}m/s}}$.” What is wrong with the student’s statement? What has the student actually described? Explain.

What is the speed of the bird in Exercise 2.4.

Acceleration is the change in velocity over time. Given this information, is acceleration a vector or a scalar quantity? Explain.

A weather forecast states that the temperature is predicted to be −5ºC the following day. Is this temperature a vector or a scalar quantity? Explain.

Give an example (but not one from the text) of a device used to measure time and identify what change in that device indicates a change in time.

There is a distinction between average speed and the magnitude of average velocity. Give an example that illustrates the difference between these two quantities.

Does a car’s odometer measure position or displacement? Does its speedometer measure speed or velocity?

If you divide the total distance travelled on a car trip (as determined by the odometer) by the time for the trip, are you calculating the average speed or the magnitude of the average velocity? Under what circumstances are these two quantities the same?

How are instantaneous velocity and instantaneous speed related to one another? How do they differ?

Is it possible for speed to be constant while acceleration is not zero? Give an example of such a situation.

Is it possible for velocity to be constant while acceleration is not zero? Explain.

Give an example in which velocity is zero yet acceleration is not.

If a subway train is moving to the left (has a negative velocity) and then comes to a stop, what is the direction of its acceleration? Is the acceleration positive or negative?

Plus and minus signs are used in one-dimensional motion to indicate direction. What is the sign of an acceleration that reduces the magnitude of a negative velocity? Of a positive velocity?

What information do you need in order to choose which equation or equations to use to solve a problem? Explain.

What is the last thing you should do when solving a problem? Explain.

What is the acceleration of a rock thrown straight upward on the way up? At the top of its flight? On the way down?

An object that is thrown straight up falls back to Earth. This is one-dimensional motion. 

(a) When is its velocity zero? 

(b) Does its velocity change direction? 

(c) Does the acceleration due to gravity have the same sign on the way up as on the way down?

Suppose you throw a rock nearly straight up at a coconut in a palm tree, and the rock misses on the way up but hits the coconut on the way down. Neglecting air resistance, how does the speed of the rock when it hits the coconut on the way down compare with what it would have been if it had hit the coconut on the way up? Is it more likely to dislodge the coconut on the way up or down? Explain.

If an object is thrown straight up and air resistance is negligible, then its speed when it returns to the starting point is the same as when it was released. If air resistance were not negligible, how would its speed upon return compare with its initial speed? How would the maximum height to which it rises be affected?

The severity of a fall depends on your speed when you strike the ground. All factors but the acceleration due to gravity being the same, how many times higher could a safe fall on the Moon be than on Earth (gravitational acceleration on the Moon is about 1/6 that of the Earth)?

How many times higher could an astronaut jump on the Moon than on Earth if his take-off speed is the same in both locations (gravitational acceleration on the Moon is about 1/6 of g on Earth)?

a) Explain how you can use the graph of position versus time in Figure 2.54 to describe the change in velocity over time. 

Identify 

(b) the time ( ta, tb , tc , td , or te ) at which the instantaneous velocity is greatest, 

(c) the time at which it is zero, and 

(d) the time at which it is negative.

                                                      Figure 2.54

(a) Sketch a graph of velocity versus time corresponding to the graph of displacement versus time given in Figure 2.55. 

(b) Identify the time or times ( ta , tb , tc , etc.) at which the instantaneous velocity is greatest. 

(c) At which times is it zero? 

(d) At which times is it negative?

(a) Explain how you can determine the acceleration over time from a velocity versus time graph such as the one in Figure 2.56. 

(b) Based on the graph, how does acceleration change over time?

(a) Sketch a graph of acceleration versus time corresponding to the graph of velocity versus time given in Figure 2.57.

(b) Identify the time or times ( ta, tb, tc , etc.) at which the acceleration is greatest.

(c) At which times is it zero? 

(d) At which times is it negative?

Consider the velocity vs. time graph of a person in an elevator shown in Figure 2.58. Suppose the elevator is initially at rest. It then accelerates for, maintains that velocity for , then decelerates for  until it stops. The acceleration for the entire trip is not constant so we cannot use the equations of motion from Motion Equations for Constant Acceleration in One Dimension for the complete trip. (We could, however, use them in the three individual sections where acceleration is a constant.) Sketch graphs of 

(a) position vs. time and 

(b) acceleration vs. time for this trip.

A cylinder is given a push and then rolls up an inclined plane. If the origin is the starting point, sketch the position, velocity, and acceleration of the cylinder vs. time as it goes up and then down the plane.

Find the following for path A in Figure 2.59: 

(a) The distance travelled. 

(b) The magnitude of the displacement from start to finish. 

(c) The displacement from start to finish.

Find the following for path B in Figure 2.59: 

(a) The distance travelled. 

(b) The magnitude of the displacement from start to finish. 

(c) The displacement from start to finish.

Find the following for path C in Figure 2.59: 

(a) The distance travelled. 

(b) The magnitude of the displacement from start to finish. 

(c) The displacement from start to finish.

Find the following for path D in Figure 2.59: 

(a) The distance travelled. 

(b) The magnitude of the displacement from start to finish. 

(c) The displacement from start to finish.

(a) Calculate Earth’s average speed relative to the Sun. 

(b) What is its average velocity over a period of one year?

A helicopter blade spins at exactly $\mathbf{100}$ revolutions per minute. Its tip is $\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{m}$ from the centre of rotation. 

(a) Calculate the average speed of the blade tip in the helicopter’s frame of reference. 

(b) What is its average velocity over one revolution?

The North American and European continents are moving apart at a rate of about $\mathbf{3}\mathbf{\text{\hspace{0.17em}cm/y}}$. At this rate how long will it take them to drift $\mathbf{500}\mathbf{km}$ farther apart than they are at present?

Land west of the San Andreas fault in southern California is moving at an average velocity of about $\mathbf{6}\mathbf{cm}/\mathbf{y}$ northwest relative to land east of the fault. Los Angeles is west of the fault and may thus someday be at the same latitude as San Francisco, which is east of the fault. How far in the future will this occur if the displacement to be made is $\mathbf{590}\mathbf{km}$ northwest, assuming the motion remains constant?