Basics of Linear Motion Physics Chapter 4

Motion Motion can be defined as a change in position relative to a stationary object. Motion is considered a relative thing because motion can only be determined by comparing an object’s position to another object. In physics, most objects in motion will be compared to the surface of Earth. But it can change depending upon what you compare it to.

Speed Galileo defined speed as the distance covered per unit of time. Speed = Distance / Time Ex. 40 m/s = 40 meters / 1 second Common units of speed are: km/hr m/s mph

Average Speed Average speed is defined as the total distance covered divided by total time interval. Average Speed = Total Distance / Total Time Average speed does not indicate instances of speeding up, or slowing down. Most of the time, average speed and instantaneous speed will NOT be the same.

Check-up A car drives 3000 miles in 6 hours. What is its speed? What is the equation you should use? Do you have to reconfigure it? Are all your units compatible? D = v t which is the same as v = d/t so 3000 miles / 6 hr is 500 miles/hr

Check-up 2 Jim travelled for 2 hours at an average speed of 48 mph and then increased his speed to 82 mph over the space of 10 seconds. What distance did he travel in that 10 seconds? d = (vi +vf) t / 2 but what is 10 seconds in hours? 10 sec (1 hr/ 3600 s) = 0.00278 hr so d = [(48 mil/hr + 82 mil/hr) /2 ] 0.00278 hr = [ 17 ] 0.00278 = 0.03726 mile

Instantaneous Speed Objects in motion do not always go a continuous speed, so they may slow down or speed up. The speed as measured at any given moment is called the instantaneous speed.

Velocity Speed is used to determine how fast an object is moving. Velocity tells us both how fast an object is moving and in what direction. Ex. 60 km/hr west Velocity is considered a vector quantity cause it gives both direction & magnitude. Speed is a scalar quantity because it only gives magnitude, and not direction.

Constant & Changing Velocity Constant velocity requires that both the speed and the direction remains the same. Therefore, constant velocity requires a constant speed in a straight line. Changing velocity implies the object either speeds up, slows down or changes directions. Changing velocity is called acceleration.

Acceleration Acceleration describes changes in motion & is defined as a change in an object’s velocity. A decrease in speed could be called a deacceleration. Acceleration is calculated by dividing the change in velocity by the time interval over which it occurred or, Acceleration = Change in Velocity / Time Interval Ex. If a car speeds up 15 km/hr over 5 seconds than, 15 km/hr / 5 s = 3 km/hr/s

Check-up 3 A car traveling at 40 mph speeds up to 60 mph over 20 seconds. What is its average acceleration? What are you trying to find? What do you know? If a = Δv / Δ t then 20 mph / 20 s = 1 mph/s

More on Acceleration Acceleration will always have 2 units of time. Acceleration can be either positive or negative. Instantaneous velocity can be calculated by multiplying acceleration by an increasingly small unit of time (more in a bit).

Galileo, Acceleration & Falling Objects Galileo first worked with acceleration by rolling objects down an inclined plane, finding the ball picks up the same amount of speed in successive seconds. Therefore, the ball accelerates at a constant rate. He discovered that the velocity change = acceleration x time. He later worked with objects in free fall, most famously by dropping cannonballs off the Leaning Tower of Pisa, assuming that things fall due to the force of gravity acting on them.

Acceleration & Inclined Planes Galileo also found that the steeper the slope the greater the rate of acceleration. He applied this to falling objects and found that all objects fall with an unchanging acceleration. He found that neglecting air resistance all objects fall at the same rate regardless of weight or size. Newton actually allowed us to calculate Instantaneous velocity using calculus. Video; explanation. So if you graph a displacement vs time graph, draw a tangent to any point on the motion curve, the slope of that line will be the instantaneous speed. Add direction & you get instantaneous velocity.

Free Fall When object falls without the effects of friction or air resistance, it is called free fall. During each second of free fall, an object will accelerate at 9.8 or 10 m/s/s, the pull of gravity. The pull of gravity is abbreviated as g. Therefore the velocity of a falling object can be found by, Velocity = gravity’s pull x time or v = gt

Check-up 4 The acceleration due to gravity is 10 m/s2. IF the lead rock climber on your rope drops a piton and it is going 7 m/s when it passes you, how fast will it be going 2 seconds later? If its speed passing you is 7 m/s, then 1 s later it is going 17 m/s and 2 s later it has a speed of 27 m/s.

Free Fall To summarize: Galileo showed that falling objects accelerate equally, regardless of mass, if air resistance is nonexistent or very small. Weight and mass always have the same ratio in the same location. Therefore, falling objects have the same acceleration because the net force on each object is only its weight and the ratio of weight to mass is always the same.

Free Fall: How Far? Galileo found that the distance a uniformly accelerating object travels is proportional to the square of time or, Distance traveled = ½ (acceleration x time x time) d = ½ gt2 Air resistance can effect a falling object in the real world, however this equation is accurate for most objects falling from a position of rest.

Check-up 5 The acceleration due to gravity (g) is 10 m/s2. IF a squirrel drops a nut from a tree & it hits the ground 3 s later, how far up was the squirrel? Use d = ½ g t2 d = ½ 10 m/s2 (3 s)(3 s) = 5 (9) = 45 m

Falling & Acceleration due to g The more air resistance an object has, the more the net force is diminished, therefore acceleration decreases. When the air resistance equals the weight of the falling object, net force is zero and the object no longer accelerates.

Falling, Terminal Velocity & Air Resistance When acceleration terminates an object has terminal speed or terminal velocity. The amt of air resistance depends upon the total surface area of the object. Paper or feathers fall much more slowly than small balls or rocks.

Necessary info to solve problems  

Sample Problem #1 While driving down the highway at constant speed v you sneeze, and your eyes close for a brief time t. How far along the highway do you travel during the sneeze?

Sample Problem #1 While driving down the highway at constant speed v you sneeze, and your eyes close for a brief time t. How far along the highway do you travel during the sneeze? d = v t

Sample Problem #2 Calculate the distance in kilometers that you travel while sneezing given that your speed is 115 km/hr and your eyes close for 0.70 seconds while sneezing? (hint: need to convert speed to meters per second or time of sneeze to hours)

Sample Problem #2 Calculate distance in km that you travel while sneezing given that your speed is 115 km/hr & your eyes close for 0.70 s while sneezing? (hint: need to convert time of sneeze to hrs) Converting s to hr: 0.70 s (1 hr / 3600 s) = 0.00019 hr …so using d = v t = 115 km/hr (0.00019 hr) = 0.02236 km

Sample Problem #3 Mala enjoys exercise and jogs a distance x at a constant speed v. What time does it take Mala to cover distance x? Calculate the time in minutes for Mala to jog 1.3 km at a constant speed of 2.0 m/s. (convert km to m first then after calculation convert s to minutes OR convert 2.0 m/s to km/min)

Sample Problem #3 Mala enjoys exercise & jogs a distance x at a constant speed v. What time does it take Mala to cover distance x? d = v t so t = d / v Calculate the time in minutes for Mala to jog 1.3 km east at a constant speed of 2.0 m/s. (convert km to m first then after calculation convert s to minutes OR convert 2.0 m/s to km/min) 1.3 km (1000 m/ 1 km) = 1300 m t = d / v = 1300 m / 2.0 m/s = 1300 m (1/2 s/m) = 650 s or 10.83 min

Sample Problem #4 A bus starting from rest uniformly accelerates along a level road. How far does the bus travel when accelerating from rest to speed vf in a time interval t? Calculate the distance traveled by the bus when it starts from rest and accelerates to 12 m/s in a time of 5 s.

Sample Problem #4 A bus starting from rest uniformly accelerates to the west along a level road. How far does the bus travel when accelerating from rest to speed vf in a time interval t? Since we want the average change in velocity, use d = (vi + vf) t / 2 Calculate the distance traveled by the bus when it starts from rest and accelerates to 12 m/s in a time of 5 s. = [0 m/s + 12 m/s] 5s = 12 m/s (5 s) = 30 m 2 2

Sample Problem #5 A cat steps off a building and splats to the ground after 12 seconds. What is its speed upon striking the ground? What was its average speed? How far did it fall?

Sample Problem #5 A cat steps off a building and splats to the ground after 12 seconds. What is its speed upon striking the ground? The cat is accelerating at 10 m/s2 (g). It takes 12 s for it to hit so 10 x 12 = 120 m/s What was its average speed? Use (vi + vf) / 2= 120 m/s / 2 = 60 m/s How far did it fall? Use d = v t or 60 m/s [12 s] = 720 m

Displacement Graphs (showing type of motion)