Finding the Linear Equation for Time and Velocity Relationship

Question

Which of the following equations represents the linear relationship between time, t, and velocity, v, shown in the table below?

t	0	1	2
v	120	152	184

Answer 1

Hello,

In this question, it is asking you to develop a generalized equation that relates velocity, v, and time, t. Velocity, by definition, is dependent upon t, so the function we are trying to solve for is v(t). The question explicitly states "the linear relationship between time, t, and velocity, v." This tells us that the relationship is linear. The general form of a linear relationship is in the general form of a linear polynomial.

Linear Polynomial General Form: ax+b

Using this, we can create an equation and equate that to v(t).

For our "a" in the general form, this is known as the "slope," or a ratio between the change in one variable to another. In this case, it's a ratio between the change in the dependent variable, v, to the independent variable, t.

The ratio can be expressed as (v₂(t) - v₁(t)) / (t₂ - t₁),

which can also be written simply as: (v₂ - v₁) / (t₂ - t₁).

Plugging in 2 points such as:

(t₁ , v₁) (t₂ , v₂ )

(0, 120) (1, 152)

By plugging these values into our ratio:

(152 - 120) / (1 - 0) = 32 / 1 = 32

To determine our "+b" from our generalized polynomial, we need to first understand what this is. This +b is the coordinate of our dependent variable, v, whenever our independent variable, t = 0. Essentially this is the v-intercept. In order to determine this, we need to set our t values equal to zero in our generalized equation, v(t) = 32 * 0 + b. However, since we have labeled our t = 0, and we know that v and t are linearly dependent, our v must have a corresponding value. On our tables, when t = 0, v(t) = 120.

120 = 32 * 0 + b

Solving for b, we get:

120 = b

We have completed every unknown piece for the equation representing the relationship between time, t, and velocity, t.

In summary:

v(t) = a * t + b

a = 32

b = 120

v(t) = 32 * t + 120

Hope this helps!

-Andrew Evans

Answer 2

Answer 3

Hello, thank you for taking the time to post your question!

The general form for a line is y = mx + b, where m is the slope and b is the y-intercept. What’s nice about this table is that the y-intercept value is given to you directly … it would be 120 because it’s the point where t = 0.

For the slope you want to pick any two points and figure out what the rate of change is between them. For this set of data it ends up being (152 – 120) / (1 – 0) = 32 / 1 = 32.

So the full equation of the line would be v = 32t + 120

Hopefully that gets you moving in the right direction! Feel free to reach out for a lesson if you have any questions beyond that! :)

Answer 4

The keyword in this whole question is linear. This means there is not going to be any exponential or parabolic relationships here it is a straight line. This means that the relationship as time passes is going to be constant or again a straight line.

you’ll notice that as each one incremental unit passes in time that velocity increases by 32 so we can put together a relationship with that.

we have an initial velocity at time zero of 120 and its time passes. It goes up by 32 for each subsequent second so we can put together the equation based on this relationship of

V = 120 + 32t

rearranging into slope intercept form it would look like

V = 32t + 120

The slope is 32 and the Y intercept is 120

If you graphed those points on a graph, it would show the slope and y-intercept as indicated above

Answer 5