Which equation can be used to solve for acceleration?
[tex]t=\frac{\Delta v}{a}[/tex]
[tex]v f=a t-v[/tex]
[tex]a=\frac{d}{t}[/tex]
[tex]\Delta v=\frac{a}{t}[/tex]
Answer (2)
Rearrange the first equation t = a Δ v to solve for a , resulting in a = t Δ v .
Rearrange the second equation v f = a t − v to solve for a , resulting in a = t v f + v .
Analyze the third equation a = t d , noting it's not a general definition of acceleration.
Rearrange the fourth equation Δ v = t a to solve for a , resulting in a = Δ v × t .
Conclude that the first equation, t = a Δ v , correctly rearranges to the standard formula for acceleration: a = t Δ v .
Explanation
Problem Analysis We need to determine which of the given equations can be used to solve for acceleration, denoted as 'a'. We will manipulate each equation to isolate 'a' on one side.
Solving the first equation Let's start with the first equation: t = a Δ v . To solve for 'a', we can multiply both sides by 'a' and then divide by 't':
t × a = Δ v
a = t Δ v
Solving the second equation Now, let's look at the second equation: v f = a t − v . To solve for 'a', we first add 'v' to both sides:
v f + v = a t
Then, we divide by 't':
a = t v f + v
Analyzing the third equation The third equation is already solved for 'a': a = t d . This equation states that acceleration is equal to the distance 'd' divided by time 't'. However, acceleration is defined as the rate of change of velocity, not the rate of change of distance. Therefore, this equation is not generally correct for acceleration.
Solving the fourth equation Finally, let's examine the fourth equation: Δ v = t a . To solve for 'a', we multiply both sides by 't':
a = Δ v × t
Final Answer Comparing the rearranged equations with the definition of acceleration, which is the change in velocity divided by the change in time ( a = t Δ v ), we see that the first equation, when rearranged, gives us the correct expression for acceleration.
Therefore, the correct equation to solve for acceleration is t = a Δ v , which rearranges to a = t Δ v .
Examples
Understanding acceleration is crucial in many real-world scenarios. For example, when designing a car, engineers need to calculate the acceleration required to reach a certain speed within a specific time frame. Similarly, in sports, athletes aim to maximize their acceleration to improve performance, whether it's a sprinter accelerating off the blocks or a basketball player driving to the basket. By using the formula a = t Δ v , we can quantify and optimize these accelerations.
The equation to solve for acceleration is a = t Δ v , which is derived from rearranging the first equation t = a Δ v . This formula accurately represents acceleration as the change in velocity over time.
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