Math Problem Statement
Given
v(t)
v
(
t
)
= 29 + 20
t
t
, where
v
v
is in
m/s
m
/
s
and
t
t
is in
s
s
, use calculus to determine the total displacement from
t
1
t
1
= 1.5
s
s
to
t
2
t
2
= 3.1
s
s
.
Solution
To find the total displacement from to given the velocity function , we need to integrate the velocity function over this time interval.
Step-by-Step Solution
-
Given Velocity Function:
-
Displacement Formula Using Calculus:
The displacement over a time interval can be calculated using the definite integral of the velocity function:
-
Set Up the Integral:
Substitute the given values for and :
-
Integrate the Velocity Function:
To integrate, apply the rules of integration to each term separately:
The integrals of the individual terms are:
Therefore, the antiderivative of is:
-
Evaluate the Integral from to :
Calculate the definite integral by evaluating the antiderivative at the upper and lower limits:
Compute and :
-
Calculate the Displacement:
So, the total displacement from to is 120 meters.
Would you like further details on this process or have any additional questions?
Here are some related questions to further explore:
- What would the displacement be if the velocity function were ?
- How does the acceleration relate to the velocity function ?
- What is the instantaneous velocity at ?
- How do you find the total distance traveled if the velocity function changes direction?
- What are the units of displacement when integrating a velocity function?
Tip: When integrating velocity to find displacement, ensure you understand the difference between displacement (net change in position) and distance traveled (total path length), especially if the velocity changes direction.
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Math Problem Analysis
Mathematical Concepts
Calculus
Integration
Formulas
Definite integral
Theorems
-
Suitable Grade Level
College
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