Solving for Distance So far, we have found the speed of an object if we are given: · how far it traveled and · how long it took to travel that far Sometimes we want to know how far something has traveled if we know: · its speed and · the amount of time it has traveled Any ideas of how we could figure out an equation to do that?
Solving for a Variable Our goal is to be able to solve any equation for any variable that appears in it. Let's look at a our formula first. The variables in this equation are s, d and t. Solving for a variable means having it alone on the left side . This equation is currently solved for "s".
Solving for a Variable Remember! · The letters are placeholders for the actual numbers that you will find or need to calculate in each problem. · Formulas can be manipulated, with the rules of mathematics, to create new formulas.
The Rules for Solving for a Variable Like in any game there are a few rules. 1. To "undo" a mathematical operation, you must do the opposite. 2. You can do anything you want (except divide by zero) to one side of an equation, as long as you do the same thing to the other. 3. If there is more than one operation going on, you must undo them in the opposite order in which you would do them, the opposite of the "order of operations." 4. You can always switch the left and right sides of an equation.
The Rules Let's solve this equation for "d" That means that when we're done we'll have d alone on the left side of the equation.
3 First, is d already alone ? If not, what is with it? A s B d C t Answer D it is already alone
4 What mathematical operation connects d and t? A d is added to t B d is multiplied by t C d is divided by t D t is subtracted from d Answer
5 What is the opposite of dividing d by t? A dividing t by d B dividing by v into t C multiplying d by t Answer D multiplying by t by d Rule 1. To "undo" a mathematical operation, you must do the opposite.
6 What must we also do if we multiply the right side by t? A divide the left side by t B multiply the left side by t C divide the left side by d Answer D divide the left side by d Rule 2. You can do anything you want (except divide by zero) to one side of an equation, as long as you do the same thing to the other.
7 Is there more than one mathematical operation acting on "d"? Yes No Answer Rule 3. If there is more than one operation going on, you must undo them in the opposite order in which you would do them, the opposite of the "order of operations."
Applying Rules 1 and 2 1. To undo a mathematical operation, you must do the opposite. 2. You can do anything you want (except divide by zero) to one side of an equation, as long as you do the same thing to the other. So we undo d being divided by t, by multiplying both sides by t. The order of s and t doesn't matter. Are we done?
Applying Rule 4 Rule 4. You can always switch the left and right sides of an equation. We've now solved our equation for d. A harder problem will to solve it for t. Before we go on to solve for t, lets use our problem solving approach for problems where we need to find d.
Solving for Distance A ball rolled at a speed of 3 m/s for 2.5 s; how far did it travel? What should we do first?
Solving for Distance A ball rolled at a speed of 3 m/s for 2.5 s; how far did it travel? Diagram Given Manipulate Find Substitute Formula Answer
Solving for Distance A ball rolled at a speed of 3 m/s for 2.5 s; how far did it travel? Diagram Given s = 3 m/s t = 2.5 s Manipulate Find d = ? Substitute Formula Answer
Solving for Distance A ball rolled at a speed of 3 m/s for 2.5 s; how far did it travel? Diagram Given s = 3 m/s s = 3 m/s s = 3 m/s t=2.5 s t =0 t = 2.5 s Manipulate Find d = ? Substitute Formula Answer
Solving for Distance A ball rolled at a speed of 3 m/s for 2.5 s; how far did it travel? Diagram Given s = 3 m/s s = 3 m/s s = 3 m/s t=2.5 s t =0 t = 2.5 s Manipulate Find d = ? Substitute Formula Answer
Solving for Distance A ball rolled at a speed of 3 m/s for 2.5 s; how far did it travel? Diagram Given s = 3 m/s s = 3 m/s s = 3 m/s t=2.5 s t =0 t = 2.5 s Manipulate Find d = ? Substitute Formula Answer
Solving for Distance A ball rolled at a speed of 3 m/s for 2.5 s; how far did it travel? Diagram Given s = 3 m/s s = 3 m/s s = 3 m/s t=2.5 s t =0 t = 2.5 s Manipulate Find d = ? Substitute Formula Answer
Solving for Distance A ball rolled at a speed of 3 m/s for 2.5 s; how far did it travel? Diagram Given s = 3 m/s s = 3 m/s s = 3 m/s t=2.5 s t =0 t = 2.5 s Manipulate Find d = ? Substitute Formula Answer d = 7.5 m
8 A car was driven at a speed of 40 m/s for 4 s; what distance did the car travel? Diagram Given Answer Manipulate Find Formula Substitute Answer
9 How far will you go if you run at a speed of 20 m/s for 6 s? Diagram Given Answer Manipulate Find Formula Substitute Answer
Finding Time So far, we have found the speed of an object if we are given how far it traveled and how long it took to travel that far. And, we have found how far an object has traveled if we know its speed and travel time. But, what if we want to know how much time it will take for an object to travel a given distance at a given speed? How would we could figure out an equation to do that?
Solving for Time Let's solve for "t" That means that when we're done, t will be on its own on the left side of the equation.
10 Is t already alone? If not, what is with it? A s B d C t D it is already alone Answer
11 What mathematical operation connects d to t? A t is being divided by d B d is being divided by t C d is being multiplied by t D t is being subtracted from d Answer
12 What is the opposite of dividing d by t? A dividing d by t B dividing s by t C multiplying d by t D multiplying t by d Answer Rule 1. To "undo" a mathematical operation, you must do the opposite.
13 What must we do if we multiply the right side by t? A divide the left side by t B multiply the left side by t C divide the left side by d Answer D divide the left side by d Rule 2. You can do anything you want (except divide by zero) to one side of an equation, as long as you do the same thing to the other.
14 Is there more than one mathematical operation acting on "d"? Yes No Answer Rule 3. If there is more than one operation going on, you must undo them in the opposite order in which you would do them, the opposite of the "order of operations."
Solving for t 1. To "undo" a mathematical operation, you must do the opposite. 2. You can do anything you want (except divide by zero) to one side of an equation, as long as you do the same thing to the other. So we undo d being divided by t, by multiplying both sides by t. or Are we done?
15 Is t on its own? If not, what is it with? A s B d Answer C t D it is already alone
16 What mathematical operation connects s to t? A t is being divided by d B t is being divided into s Answer C t is being multiplied by s D t is being subtracted from s
17 What is the opposite of multiplying t by s? A dividing t by s B dividing t by t Answer C multiplying t by t D multiplying t by s
Solving for t 1. To "undo" a mathematical operation, you must do the opposite. 2. You can do anything you want (except divide by zero) to one side of an equation, as long as you do the same thing to the other.
18 Is t alone on the left? If not, what is it with? A s B d C t D it is alone Answer
Solving for Time A ball rolls at a speed of 2 m/s; How much time will it take the ball to travel 9 m?
Solving for Time A ball rolls at a speed of 2 m/s; How much time will it take the ball to travel 9 m? Diagram Given Manipulate Find Formula Substitute Answer
Solving for Time A ball rolls at a speed of 2 m/s; How much time will it take the ball to travel 9 m? Diagram Given s = 2 m/s s = 2 m/s s = 2 m/s d = 9 m d = 9 m Manipulate Find t = ? Formula Substitute Answer
Solving for Time A ball rolls at a speed of 2 m/s; How much time will it take the ball to travel 9 m? Diagram Given s = 2 m/s s = 2 m/s s = 2 m/s d = 9 m d = 9 m Manipulate Find t = ? Formula Substitute Answer t = 4.5 s
19 How much time will it take you to travel a distance of 120 m if you can move at a constant speed of 20 m/s? Diagram Given Answer Manipulate Find Formula Substitute Answer
20 How much time will it take you to travel a distance of 150 m at a speed of 30 m/s? Diagram Given Answer Manipulate Find Formula Substitute Answer
Why do Algebra to Rearrange the Formula? Why not just plug numbers into the original formula? Suppose you wanted to know how far you can travel at different speeds for different times. speed [mph] time [hours] distance [miles] 30 1 30 2 45 3 60 2 Would you rather... · plug the numbers into and solve for d 4 times? -OR- · plug numbers 4 times into the new formula you derive once?
Why do Algebra to Rearrange the Formula? Why not just plug numbers into the original formula? Suppose you wanted to know how much time it takes you to travel at different distances at different speeds? distance [miles] speed [mph] time [hours] 200 40 200 50 300 50 300 60 Would you rather... · plug the numbers into and solve for d 4 times? -OR- · plug numbers 4 times into the new formula you derive once?
21 A truck was driven at a speed of 35 m/s for 12 s; what distance did the car travel? Diagram Given Answer Manipulate Find Formula Substitute Answer
22 An boat sails 800 m in 120 s; how fast is the boat moving? Diagram Given Answer Manipulate Find Formula Substitute Answer
23 How much time will it take a airplane to travel 2000 m if it can maintain a constant speed of 160 m/s? Diagram Given Answer Manipulate Find Formula Substitute Answer
24 How far will a tortoise go in 240 s if it can aintain a constant speed of 0.1 m/s? Diagram Given Answer Manipulate Find Formula Substitute Answer
25 A wildcat sprinted a distance of 60 meters in 8 s; what was the speed of the wildcat? Diagram Given Manipulate Answer Find Formula Substitute Answer
26 If a motorcycle can travels at a constant 40 m/s, how much time will it take the motorcycle to travel 4200 m? Diagram Given Manipulate Answer Find Formula Substitute Answer
Another Perspective Click here for a "minute physics" video on measuring time and distance.
Constant Speed Return to Table of Contents
Constant Speed We have just figured out how fast something moves over a given distance and time. How would you know that the object was traveling the same speed from start to finish? — that is, at a Constant Speed?
Constant Speed Suppose two cars leak oil leaving one drop on the road every second... What can you tell about the motion of the two cars as they go from 0 to 20 meters? Which car is the fastest?.............Why?
Constant Speed Suppose two cars leak oil leaving one drop on the road every second... Which car is traveling at a constant speed? What is the characteristic of constant speed?
27 A car travels at a constant speed of 10 m/s. This means the car: A increases its speed by 10 meters every second. B decreases its speed by 10 meters every second. Answer C moves with an acceleration of 10 meters every second. D moves 10 meters every second.
28 Which vehicle is the fastest? Answer A the truck B the car C the bus D the bicycle
29 Which vehicle is traveling at a constant speed over the 20 m traveled? Answer A the truck B the car C the bus D the bicycle
Distance versus Time Graphs Return to Table of Contents
Graphing Motion So far we have looked at motion using a number line to indicate the distance traveled and the number of dots to indicate the time. t=0[s] t=1[s] t=2[s] t=10[s] Another way to look at motion is to make a graph plotting the distance (along the y-axis) against time (along the x-axis).
Graphing Motion First, we'll make a table from our observations. t=0[s] t=1[s] t=2[s] t=10[s] t (s) 0 1 2 3 4 5 6 7 8 9 10 d (m) 0 2 4 6 8 10 12 14 16 18 20 Now let's graph the data in the table.
Graphing Motion 20 18 16 14 12 10 8 6 4 2 0 0 2 4 6 8 10 12 14 16 18 20 What would you do differently in this graph?
Graphing Motion d(m) 20 18 16 14 12 10 8 6 4 2 0 0 2 4 6 8 10 12 14 16 18 20 t (s) What would you do differently in this graph?
Graphing Motion d(m) 20 18 16 14 12 10 8 6 4 2 0 0 2 4 6 8 10 12 14 16 18 20 t (s) What would you do differently in this graph?
Graphing Motion d(m) 20 16 12 8 4 0 0 2 4 6 8 10 t (s) What would you do differently in this graph?
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