A car speedometer measures only speed, since it gives no indication of the direction in which the car is traveling.

Does a car speedometer measure speed, velocity, or both?

If the velocity of an object is constant, the speed must also be constant. (A constant velocity means that the speed and direction are both constant.) If the speed of an object is constant, the velocity CAN vary. For example, a car traveling around a curve at constant speed has a varying velocity, since the direction of the velocity vector is changing.

Can an object have a varying speed if its velocity is constant? Can it have varying velocity if its speed is

constant? If yes, give examples in each case.

constant? If yes, give examples in each case.

When an object moves with constant velocity, the average velocity and the instantaneous velocity are the same at all times.

When an object moves with constant velocity, does its

average velocity during any time interval differ from its

instantaneous velocity at any instant?

average velocity during any time interval differ from its

instantaneous velocity at any instant?

No, if one object has a greater speed than a second object, it does not necessarily have a greater acceleration. For example, consider a speeding car, traveling at constant velocity, which passes a stopped police car. The police car will accelerate from rest to try to catch the speeder. The speeding car has a greater speed than the police car (at least initially!), but has zero acceleration. The police car will have an initial speed of zero, but a large acceleration.

If one object has a greater speed than a second object, does the first necessarily have a greater acceleration? Explain,

using examples.

using examples.

The accelerations of the motorcycle and the bicycle are the same, assuming that both objects travel in a straight line. Acceleration is the change in velocity divided by the change in time. The magnitude of the change in velocity in each case is the same, 10 km/h, so over the same time interval the accelerations will be equal.

Compare the acceleration of a motorcycle that accelerates

from 80 km/h to 90 km/h with the acceleration of a bicycle

that accelerates from rest to 10km/h in the same time.

from 80 km/h to 90 km/h with the acceleration of a bicycle

that accelerates from rest to 10km/h in the same time.

Yes, for example, a car that is traveling northward and slowing down has a northward velocity and a southward acceleration.

Can an object have a northward velocity and a southward

acceleration? Explain.

acceleration? Explain.

Yes. If the velocity and the acceleration have different signs (opposite directions), then the object is slowing down. For example, a ball thrown upward has a positive velocity and a negative acceleration while it is going up. A car traveling in the negative x-direction and braking has a negative velocity and a positive acceleration.

Can the velocity of an object be negative when its acceleration is positive? What about vice versa?

Both velocity and acceleration are negative in the case of a car traveling in the negative x-direction and speeding up. If the upward direction is chosen as +y, a falling object has negative velocity and negative acceleration.

Give an example where both the velocity and acceleration

are negative.

are negative.

Car A is going faster at this instant and is covering more distance per unit time, so car A is passing car B. (Car B is accelerating faster and will eventually overtake car A.)

Two cars emerge side by side from a tunnel. Car A is trav

eling with a speed of 60km/h and has an acceleration of 40km/h/min. Car B has a speed of 40km/h and has an acceleration of 60 km/h/min. Which car is passing the other as they come out of the tunnel? Explain your reasoning.

eling with a speed of 60km/h and has an acceleration of 40km/h/min. Car B has a speed of 40km/h and has an acceleration of 60 km/h/min. Which car is passing the other as they come out of the tunnel? Explain your reasoning.

Yes. Remember that acceleration is a change in velocity per unit time, or a rate of change in velocity. So, velocity can be increasing while the rate of increase goes down. For example, suppose a car is traveling at 40 km/h and a second later is going 50 km/h. One second after that, the car’s speed is 55 km/h. The car’s speed was increasing the entire time, but its acceleration in the second time interval was lower than in the first time interval.

Can an object be increasing in speed as its acceleration decreases? If so, give an example. If not, explain.

If there were no air resistance, the ball’s only acceleration during flight would be the acceleration due to gravity, so the ball would land in the catcher’s mitt with the same speed it had when it left the bat, 120 km/h. The path of the ball as it rises and then falls would be symmetric.

A baseball player hits a ball straight up into the air. It leaves the bat with a speed of 120km/h. In the absence of air resistance, how fast would the ball be traveling when the catcher catches it?

(a)

If air resistance is negligible, the acceleration of a freely falling object stays the same as the object falls toward the ground. (Note that the object’s speed increases, but since it increases at a constant rate, the acceleration is constant.)

(b) In the presence of air resistance, the acceleration decreases. (Air resistance increases as speed increases. If the object falls far enough, the acceleration will go to zero and the velocity will become constant. See Section 5-6.)

If air resistance is negligible, the acceleration of a freely falling object stays the same as the object falls toward the ground. (Note that the object’s speed increases, but since it increases at a constant rate, the acceleration is constant.)

(b) In the presence of air resistance, the acceleration decreases. (Air resistance increases as speed increases. If the object falls far enough, the acceleration will go to zero and the velocity will become constant. See Section 5-6.)

As a freely falling object speeds up, what is happening to its acceleration—does it increase, decrease, or stay the same? (a) Ignore air resistance, (b) Consider air resistance.

Average speed is the displacement divided by the time. If the distances from A to B and from B to C are equal, then you spend more time traveling at 70 km/h than at 90 km/h, so your average speed should be less than 80 km/h. If the distance from A to B (or B to C) is x, then the total distance traveled is 2x. The total time required to travel this distance is x/70 plus x/90

You travel from point A to point B in a car moving at a constant speed of 70km/h. Then you travel the same distance from point B to another point C, moving at a constant speed of 90km/h. Is your average speed for the entire trip from A to C 80 km/h? Explain why or why not.

Yes. For example, a rock thrown straight up in the air has a constant, nonzero acceleration due to gravity for its entire flight. However, at the highest point it momentarily has a zero velocity. A car, at the moment it starts moving from rest, has zero velocity and nonzero acceleration.

Can an object have zero velocity and nonzero acceleration at the same time? Give examples.

Yes. Anytime the velocity is constant, the acceleration is zero. For example, a car traveling at a constant 90 km/h in a straight line has nonzero velocity and zero acceleration.

Can an object have zero acceleration and nonzero velocity at the same time? Give examples.

A rock falling from a cliff has a constant acceleration IF we neglect air resistance. An elevator moving from the second floor to the fifth floor making stops along the way does NOT have a constant acceleration. Its acceleration will change in magnitude and direction as the elevator starts and stops. The dish resting on a table has a constant acceleration (zero).

Which of these motions is not at constant acceleration: a rock falling from a cliff, an elevator moving from the second floor to the fifth floor making stops along the way, a dish resting on a table?

The time between clinks gets smaller and smaller. The bolts all start from rest and all have the same acceleration, so at any moment in time, they will all have the same speed. However, they have different distances to travel in reaching the floor and therefore will be falling for different lengths of time. The later a bolt hits, the longer it has been accelerating and therefore the faster it is moving. The time intervals between impacts decrease since the higher a bolt is on the string, the faster it is moving as it reaches the floor. In order for the clinks to occur at equal time intervals, the higher the bolt, the further it must be tied from its neighbor. Can you guess the ratio of lengths?

In a lecture demonstration, a 3.0-m-long vertical string with ten bolts tied to it at equal intervals is dropped from the ceiling of the lecture hall. The string falls on a tin plate, and the class hears the clink of each bolt as it hits the plate. The sounds will not occur at equal time intervals. Why? Will the time between clinks increase or decrease near the end of the fall? How could the bolts be tied so that the clinks occur at equal intervals?