Its important for us to know the quantities for motion for scalars and vectors. As we talked about before scalars are just magnitude, where as, vectors are magnitude and direction. In 1-D motion the scalars we will be using include distance and speed. Where as, the vectors will be displacement, velocity and acceleration. So here are some important definitions for each of these.
- Distance: how far apart two points in space are from another ( ie 5 m)
- Displacement: the relative location of one point in space to another ( ie 5 m east)
- Speed: how fast an object is moving (ie 35 m/s)
- Velocity: How fast and in what direction an object is traveling (ie 35 m/s upwards)
- Acceleration: the change of an object’s velocity with time (ie 45 m/s^2)
On the coordinate system, we can use the x and y axis as a reference point for direction.
Lets take a closer look at a pair that get confused often. Speed and Velocity.
Speed is how fast an object is moving at that instant and to find the average speed you divide total distance traveled by total time. However, this can sometimes be wrong. Say you are driving on a curvy road. You will change speed during turns more or less. But according to the average speed equation it only takes into account distance and time.
Velocity on the other hand is how fast and in what direction an object is moving. And to find the average you take the total displacement over the total time. So it will take into account the curves and changes.
So lets do an example:
A foot ball player runs from his own goal line to the opposing team’s goal line, returning to his thirty yard line, all in 25.5 s. Calculate his average speed and magnitude of his average velocity in yards / s
We know average speed is total distance traveled divided by time.
For average velocity it is total displacement divided by total time.
As you can see these are two very different answers. So it is important to make sure you know what is actually being asked.
Graphical Interpretation of velcoity
When graphing velocity make sure the time is on the x axis and displacement is on the y axis. If you do this, the slope of the graph between any two points is the average velocity in that time interval. As two points become closer together the value of the average will be closer to the instantaneous velocity at either point.
Lets do another example:
A tennis player moves in a straight line path as shown in the figure below. Find her average velocity in the time intervals (a) 0 to 1.0 s (b) 0 to 4.0 s (c) 1.0 s to 5.0 s and (d) 0 to 5.0 s
Acceleration is defined at the time rate of change of the velocity. But what does that mean? The best way to explain it is how much more force you are adding into the forward motion. This can also be negative and work by slowing down the motion.
For 1 Dimensional motion for acceleration we have this equation:
v = v0 + (a)(t)
If acceleration is constant then:
Three vary important equations to know for kinematics and projectile motion.
|Velocity as a function of time||v= v0 + at|
|Displacement as a function of time||Delta (D) X= v0t + 1/2 at2|
|Velocity as a function of displacement||v2 = v0 2 + 2a (D)x|
So lets work some homework problems out!
- The speed of a nerve impulse in the human body is about 100 m/s. If you accidentally stub you toe in the dark, estimate the time it takes the nerve impulse to travel to your brain. (Assume that you are approximately 1.60 m tall and that the nerve impulse travels at uniform speed.)
We know the speed equation which is speed = distance / time. First we move time by itself and covert the equation into time = distance / speed. Plug in the numbers. Meters cancel out. and the answer is 0.16 s
2. A football player runs from his own goal line to the opposing team’s goal line, returning to his twenty-yard line, all in 23.4 s. Calculate his average speed and the magnitude of his average velocity. (Enter your answers in yards/s.) (a) Calculate his average speed. (b) calculate the magnitude of his average velocity.
This question is just like the one before and we will follow the same steps.
3. To qualify for the finals in a racing event, a race car must achieve an average speed of 255 km/h on a track with a total length of 2,400 m. If a particular car covers the first half of the track at an average speed of 204 km/h, what minimum average speed must it have in the second half of the event in order to qualify?
First we are gonna start with total time equaling total distance over the required ave speed to get the time
Then the time spent for the first half of the trip
Now find the time for the second half of the trip and then the required ave speed
4. A paper in the journal Current Biology tells of some jellyfish-like animals that attack their prey by launching stinging cells in one of the animal kingdom’s fastest movements. High-speed photography showed the cells were accelerated from rest for 700 ns at 5.30 ✕ 107 m/s2. Calculate the maximum speed reached by the cells and the distance traveled during the acceleration.
(a)Calculate the maximum speed (in m/s) reached by the cells.
(b)Calculate the distance (in m) traveled during the acceleration.
5. A graph of position versus time for a certain particle moving along the x-axis is shown in the figure below. Find the instantaneous velocity at the following instants.
(a) t = 1.00 s
(b) t = 3.00 s
(c) t = 4.50 s
(d) t = 7.50 s
6. Runner A is initially 5.0 mi west of a flagpole and is running with a constant velocity of 8.0 mi/h due east. Runner B is initially 1.0 mi east of the flagpole and is running with a constant velocity of 3.0 mi/h due west. How far are the runners from the flagpole when they meet?
7. A particle starts from rest and accelerates as shown in the figure below.
(a) Determine the particle’s speed at t = 10.0 s and at t = 20.0 s.
|t = 10.0 s|
|t = 20.0 s|
(b) Determine the distance traveled in the first 20.0 s. (Enter your answer to one decimal place.)
8. A 50.0-g Super Ball traveling at 27.5 m/s bounces off a brick wall and rebounds at 17.5 m/s. A high-speed camera records this event. If the ball is in contact with the wall for 4.25 ms, what is the magnitude of the average acceleration of the ball during this time interval?
9. The average person passes out at an acceleration of 7g (that is, seven times the gravitational acceleration on Earth). Suppose a car is designed to accelerate at this rate. How much time would be required for the car to accelerate from rest to 53.7 miles per hour? (The car would need rocket boosters!)
10. A truck covers 40.0 m in 7.05 s while uniformly slowing down to a final velocity of 3.45 m/s. (a) Find the truck’s original speed. (b) Find it’s accerleration
11. A certain aircraft has a liftoff speed of 118 km/h.(a) What minimum constant acceleration does the aircraft require if it is to be airborne after a takeoff run of 233 m? (b) How long does it take the aircraft to become airborne?
12. A jet plane lands with a speed of 112 m/s and can accelerate at a maximum rate of −4.90 m/s2 as it comes to rest. (a) From the instant the plane touches the runway, what is the minimum time needed before it can come to rest? (b) Can this plane land on a small tropical island airport where the runway is 0.800 km long?
13. A train is traveling down a straight track at 27 m/s when the engineer applies the brakes, resulting in an acceleration of −1.0 m/s2 as long as the train is in motion. How far does the train move during a 54-s time interval starting at the instant the brakes are applied?
14. A hockey player is standing on his skates on a frozen pond when an opposing player, moving with a uniform speed of 2.0 m/s, skates by with the puck. After 1.80 s, the first player makes up his mind to chase his opponent. If he accelerates uniformly at 0.30 m/s2, determine each of the following. (a) How long does it take him to catch his opponent? (Assume the player with the puck remains in motion at constant speed.) (b) How far has he traveled in that time?
15. A ball is thrown vertically upward with a speed of 14.0 m/s. (a) How high does it rise? (b) How long does it take to reach its highest point? (c) How long does the ball take to hit the ground after it reaches its highest point? (d) What is its velocity when it returns to the level from which it started?
16. A ball is thrown directly downward with an initial speed of 7.75 m/s, from a height of 30.5 m. After what time interval does it strike the ground?
17. Traumatic brain injury such as concussion results when the head undergoes a very large acceleration. Generally, an acceleration less than 800 m/s2 lasting for any length of time will not cause injury, whereas an acceleration greater than 1,000 m/s2 lasting for at least 1 ms will cause injury. Suppose a small child rolls off a bed that is 0.41 m above the floor. If the floor is hardwood, the child’s head is brought to rest in approximately 2.1 mm. If the floor is carpeted, this stopping distance is increased to about 1.4 cm. Calculate the magnitude and duration of the deceleration in both cases, to determine the risk of injury. Assume the child remains horizontal during the fall to the floor. Note that a more complicated fall could result in a head velocity greater or less than the speed you calculate. (a) hardwood floor magnitude (b) hardwood floor duration (c) carpeted floor magnitude (d) carpeted floor duration
18. A tennis player tosses a tennis ball straight up and then catches it after 1.29 s at the same height as the point of release. (a) What is the acceleration of the ball while it is in flight? (b) What is the velocity of the ball when it reaches its maximum height? (c) Find the initial velocity of the ball. (d) Find the maximum height it reaches.
19. A baseball is hit so that it travels straight upward after being struck by the bat. A fan observes that it takes 2.90 s for the ball to reach its maximum height. (a) Find the ball’s initial velocity. (b) Find the height it reaches
20. A mouse is eating cheese 3.54 meters from a sleeping cat. When the cat wakes up, the mouse immediately starts running away from the cat with a constant velocity of 1.79 m/s. The cat immediately starts chasing the mouse with a constant velocity of 3.63 m/s. Assume the cat and the mouse start running instantaneously so there are no accelerations to worry about. (a) How many seconds after they begin running does the cat catch the mouse? (b) How far does the cat have to run to catch the mouse?
Please bear with me while I make videos for the questions that do not have them. Also here is a link to work out more problems on Numerade