a Construct a two-column proof. Start with stating the given information.
B
b Use the formula that you have proven in Part A to find acceleration.
A
aTwo-Column Proof:
Statements
Reasons
1.
d=vt+1/2at^2
1.
Given
2.
d-vt=vt-vt+1/2at^2
2.
Subtraction Property of Equality
3.
d-vt=1/2at^2
3.
Substitution Property of Equality
4.
2(d-vt)=2(1/2at^2)
4.
Multiplication Property of Equality
5.
2(d-vt)=at^2
5.
Substitution Property of Equality
6.
2d-2vt=at^2
6.
Distributive Property
7.
2d-2vt/t^2=at^2/t^2
7.
Division Property of Equality
8.
2d-2vt/t^2=a
8.
Substitution Property of Equality
9.
a=2d-2vt/t^2
9.
Symmetric Property of Equality
B
bProperty: Substitution Property of Equality
Acceleration: 3 ft/s^2
Practice makes perfect
a The relation between acceleration a, distance d, velocity v, and time t is represented by the formula given below.
d=vt+1/2at^2
We will prove that if the values for distance, velocity, and time are known, then the acceleration of an object can be found by the given formula.
a=2d-2vt/t^2
In order to prove that, we will isolate a in the given equation by using the Properties of Equality. First, we will state the given to construct a two-column proof.
Given
d=vt+1/2at^2
Then, we will subtract vt from both sides of the equation by the Subtraction Property of Equality.
Subtraction Property of Equality
d- vt=vt- vt+1/2at^2
Next, we will simplify the terms by using the Substitution Property of Equality.
Substitution Property of Equality
d-vt=1/2at^2
For the next step, we will multiply both sides of the equation by 2. This step is justified by the Multiplication Property of Equality.
Multiplication Property of Equality
2(d-vt)= 2(1/2at^2)
After multiplying the terms, we will simplify the equation and use the Substitution Property of Equality.
Substitution Property of Equality
2(d-vt)=at^2
Now, we will distribute 2 by the Distribution Property of Equality.
Distribution Property of Equality
2d- 2vt=at^2
For the seventh step, we need to divide both sides of the equation by t^2, which is justified by the Division Property of Equality.
Division Property of Equality
2d-2vt/t^2=at^2/t^2
Before the last step, we will again simplify the terms and use the Substitution Property of Equality.
Substitution Property of Equality
2d-2vt/t^2=a
As a final step, we will exchange the left-hand side and the right-hand side by the Symmetric Property of Equality.
Symmetric Property of Equality
a=2d-2vt/t^2
Combining these steps, we will construct two-column proof.
Statements
Reasons
1.
d=vt+1/2at^2
1.
Given
2.
d-vt=vt-vt+1/2at^2
2.
Subtraction Property of Equality
3.
d-vt=1/2at^2
3.
Substitution Property of Equality
4.
2(d-vt)=2(1/2at^2)
4.
Multiplication Property of Equality
5.
2(d-vt)=at^2
5.
Substitution Property of Equality
6.
2d-2vt=at^2
6.
Distribution Property of Equality
7.
2d-2vt/t^2=at^2/t^2
7.
Division Property of Equality
8.
2d-2vt/t^2=a
8.
Substitution Property of Equality
9.
a=2d-2vt/t^2
9.
Symmetric Property of Equality
b Let's find the acceleration of an object if it has d=2850 feet, t=30 seconds, and v=50 feet per second. To do that, we will use the Substitution Property of Equality and substitute the given values into the formula.
