Physics 20
Work Booklet
Unit I – Kinematics
Name: ______
Scalars and vectors
1. Draw a seating plan using the statements below.
(a) Chad is 2.0 m [left] of Dolores.
(b) Ed is 4.5 m [right] of Chad.
(c) Greg is 7.5 m [left] of Chad.
(d) Hannah is 1.0 m [right] of Ed.
(e) What is the displacement of a teacher who walks from Greg to Hannah?
2. While parking his new trailer, Al first moves the trailer 10.0 m [N], then 3.0 m [S], and then finally 18.0 m [N]. His total displacement while parking is ______m [N].
3. At soccer practice, players are required to increase their endurance by performing sprints. The coach sets up the following pattern for the players to do at the end of practice.
Each player begins at the goal line and the lines are 10 m apart. The total distance that each player will sprint is:
4. An athlete at the track club trains for the 100-m race. At 0 s, he is at the starting line. At 1.0 s, he reaches the 10-m line. At 2.0 s, he reaches the 20-m line. At 3.0 s, he reaches the 30-m line. The magnitude of the runner’s displacement at 3.0 s relative to the starting point is:
5. A motorcycle travels 80 m [north] in 15 s. The motorcycle then turns around and travels 60 m [south] in 10 s. During the 25-s interval, the motorcycle
a. travels a distance of:
b. has an average speed of:
c. has an average velocity of:
Answers: 1.e) 13.0 m [right]; 2. 25 m [N]; 3. 1.2 x 102 m; 4. 30m; 5. a)140 m, b) 5.6m/s, c) 0.8 m/s [N]
Position-Time Graphs
1.
a) 8.0 s b) 12 s c) 1.0 s
2. Two rollerbladers, A and B, are having a race. B gives A a head start of 5.0 s. Each rollerblader moves with a constant velocity. Assume that the time taken to reach constant velocity is negligible. If A travels 100.0 m [right] in 20.0 s and B travels 112.5 m [right] in 15.0 s,
(a) graph the motions of both rollerbladers on the same graph.
(b) find the time, position, and displacement at which B catches up with A.
3. Sketch a position-time graph for a bear starting 1.2 m from a reference point, walking slowly away at constant velocity for 3.0 s, stopping for 5.0 s, backing up at half the speed for 2.0 s, and finally stopping.
4. Complete the instantaneous velocity-time data table for the graph below.
5.
Answers: 1. 0.67 m/s [E]; 2.b) time= 15.0 s, position and displacement= 75.0 m [right] 4. 3.8, 7.0, 0.0, -7.5 5. a)5.0 m/s, b)5.0 m [N], c)1.3 m/s [S], d)0.98 m/s, e)0.33 m/s [N]
Acceleration
Acceleration is defined as the rate of change of velocity of an object. Acceleration can of course change, but in physics 20 we will deal with average or constant accelerations.
The symbol for average or constant acceleration is . The units for acceleration are m/s2. Acceleration is a vector because velocity is a vector.
The following equations are all valid whenever an object is accelerating at a constant (or average) rate. The one you will use depends on what data is given to you in the question.
When an object is slowing down as it rises or speeding up as it falls (because of gravity), the acceleration of the object will always be –9.81 m/s2. In general, an acceleration can be negative it the object speeds up in the negative direction or slows down in the positive direction.
Example
A car traveling at 10 m/s accelerates to get past another car. If it speeds up to a velocity of 25 m/s in a time of 6.0 seconds, what is the acceleration of the car?
Example
How fart has the car in the above example traveled during its time of acceleration?
Example
A rock falls from a height of 50 m above the ground. What is its speed when it strikes the ground?
Example
A ball rolling down a hill accelerates at a rate of 3.0 m/s2. If it initially moves at a speed of 5.0 m/s, how far will it travel in 10 seconds?
Velocity-Time Graphs
A velocity-time graph always has velocity on the y-axis and time on the x-axis. It can take any one (or combination) of the following shapes:
constant constant constant increasing decreasing
acceleration velocity negative acceleration acceleration
acceleration
The instantaneous velocity of an object at can be directly read off of the graph. Change in velocity, average velocity, acceleration, and displacement can be calculated from the graph.
Calculating Change in Velocity From a Velocity-Time Graph
The change in velocity of an object during an interval of time can be calculated the following equation:
Simply read the velocity of the object off the graph at two times and then subtract them to get the change in velocity.
Example
What is the change in velocity of the object in the following graph between t= 0 seconds and t= 10 seconds?
Calculating Acceleration From a Velocity-Time Graph
Recall that acceleration is defined as the rate of change of velocity of an object. The average acceleration of an object during an interval of time can therefore be calculated using the basic acceleration equation:
Find the change in velocity during an interval of time, as above, and divide it by the time interval. Note that this is really just finding the slope of the graph (if the graph is a straight line). Also note that if the graph is perfectly straight, then the acceleration solved for will be a constant acceleration rather than an average acceleration.
Example
What is the acceleration of the object in the previous graph?
Example
What is the average acceleration of the object traveling West in the graph below?
Calculating Displacement From a Velocity-Time Graph
Displacement of an object can be calculated from a velocity-time graph by calculating the total area between the x-axis and the graph line. If the graph line is above the x-axis, the displacement will be considered to be positive. If the graph line is below the x-axis, the displacement will be considered to be negative.
Sometimes the area must be broken up into sections in order to find the total area. If this is the case, each section will either be a rectangle or a triangle (for the problems that you will see in physics 20). The equation for the area of a rectangle is as follows:
The equation for the area of a triangle is as follows:
Example
What is the displacement of the object in the following graph?
Accleration-Time Graphs
An acceleration-time graph always has acceleration on the y-axis and time on the x-axis. It can take any one (or combination) of the following shapes:
constant object constant
velocity stopped negative
velocity
The instantaneous acceleration of an object at can be directly read off of the graph. Change in acceleration of an object can be calculated by taking two readings for acceleration then subtracting the initial value from the final value.
Two-Dimensional Vectors
If a vector is in a direction other than x (horizontal) or y (vertical) (in other words, it has a combination of both), then we typically need to break the vector up into its x-components and its y-components. We will likely perform some kind of operations with both components then re-combine them near the end of the problem.
Finding the X-Component of a Vector
If the entire vector is horizontal, the x-component of the vector is the entire vector. Remember to include a negative sign if the vector is pointing to the left. If the entire vector is vertical, the x-component of the vector is zero.
If, however, the vector is partly horizontal and partly vertical, a triangle should be drawn to find the x-component.
q
adjacent hypotenuse (the vector itself)
opposite
The vector will usually be the hypotenuse of the triangle. The x-component of the vector will either be the adjacent or opposite side of the triangle, depending on which corner the angle is measured from.
If you need to find the opposite side of the triangle in order to find the x-component, use the cosine function:
If you need to find the adjacent side of the triangle in order to find the x-component, use the sine function:
Example
A man walks 200 m [300 S of E]. What is the x-component of the man’s displacement?
Finding the Y-Component of a Vector
If the entire vector is vertical, the y-component of the vector is the entire vector. Remember to include a negative sign if the vector is pointing to the down. If the entire vector is horizontal, the y-component of the vector is zero.
If, however, the vector is partly horizontal and partly vertical, a triangle should be drawn to find the y-component.
q
adjacent hypotenuse (the vector itself)
opposite
The vector will usually be the hypotenuse of the triangle. The y-component of the vector will either be the adjacent or opposite side of the triangle, depending on which corner the angle is measured from.
If you need to find the opposite side of the triangle in order to find the y-component, use the cosine function:
If you need to find the adjacent side of the triangle in order to find the y-component, use the sine function:
Example
A man walks 200 m [300 S of E]. What is the y-component of the man’s displacement?
Two-Dimensional Vector Addition
The normal rules of addition do not apply to vectors that are in two-dimensions. For instance, while 3 plus 4 normally equals 7, it will equal 5 if the vectors are at right angles to one another. To add vectors that are not all completely horizontal or are not all completely vertical, you should complete the following steps:
1. Draw a vector diagram showing all of the vectors. It doesn’t matter what order the vectors are drawn in, but the head of one vector should always be drawn to the tail of the next. The exception to this rule is the resultant vector (the sum of all the vectors), which is drawn from the tail of the first vector to the head of the last vector (i.e. drawn from start to finish). The resultant vector is typically drawn as a dotted line.
2. Find the x-components of all the vectors. Add them up.
3. Find the y-components of all the vectors. Add them up.
4. Re-draw a vector diagram with the two vectors you found in steps 2 and 3.
5. Use the Pythagorean Theorem and trigonometry to find the magnitude and direction of the resultant vector. The resultant vector is the sum that you were looking for.
Example
A student walks 100 m East, then 200 m [300 N of E], then 500 m South.
a. Find the resultant vector.
b. What is the sum of these vectors?
c. What is the student’s total displacement?
Relative Motion and Navigation
Motion always needs to be defined relative to something else, whether that “something else” is expressly stated or not. For instance, when we talk of an airplane traveling at 800 km/h East, we typically mean that the airplane is traveling at 800 km/h East relative to the Earth. When we talk of a baseball traveling at 160 km/h East, we typically mean that the baseball is traveling at 800 km/h East relative to the batter.
But what if a baseball was thrown at 160 km/h East inside and airplane that was traveling at 800 km/h East? The velocity of the airplane relative to the Earth would still be 800 km/h East. The velocity of the ball relative to the airplane would be 160 km/s East. The velocity of the ball relative to the Earth, however, would be 960 km/h. If the ball was traveling West, the velocity of the ball relative to the Earth would be 640 km/h East. Clearly, the velocity (and motion in general) depends on what it is defined relative to.
Two-dimensional relative motion problems typically come in the form of navigation problems.
Solving Navigation Problems
1. Draw a vector diagram. Use the following guide to help you decide which vectors should be drawn as “solid line vectors” and which should be drawn as “dotted line vectors” (the resultant vector):
a. If a pilot or captain aims in a certain direction or heads in a certain direction, draw the vector as a solid line vector.
b. If the pilot or captain actually goes in a certain direction or wants to go in a certain direction, or if you’re dealing with the resultant velocity, draw the vector as a dotted line vector.
Remember that a solid line vector (resultant vector) should always be drawn head to tail, while dotted line vectors should always be drawn tail to head.
2. Using the Pythagorean Theorem and trigonometry, solve for the unknown vector.
Example
The captain of a ship aims his ship due North at a speed of 10 m/s. A current in the ocean of 2.0 m/s [West] carries the ship off course. Relative to the shore, what is the ship’s resultant velocity?
Example
The pilot of a plane wants to head due North at a speed of 200 m/s but a wind of 20 m/s blows to the west. At what velocity and at what angle must the pilot aim the plane?
Projectile Motion
When any object is thrown into the air, it becomes a projectile. When we study the motion of an object that is thrown into the air, we call it projectile motion.
When an object is thrown straight up into the air, it comes straight back down. We have analyzed problems like this before using the four acceleration equations. But what if the object is thrown at an angle other than straight up (or straight down)?