Which equation can be used to solve for acceleration?
[tex]t=\frac{\Delta v}{a}[/tex]
[tex]v f=a t-v i[/tex]
[tex]a=\frac{d}{t}[/tex]
[tex]\Delta v=\frac{a}{t}[/tex]
Answer (2)
Analyze each equation to isolate 'a'.
Equation 1: a = t Δ v .
Equation 2: a = t v f + v i .
Equation 3 is not a general definition of acceleration.
Equation 4 is incorrect.
The equations that can be used to solve for acceleration are t = a Δ v and v f = a t − v i .
Explanation
Problem Analysis We are given four equations and asked to identify which one can be used to solve for acceleration, denoted as 'a'. To do this, we will examine each equation and rearrange it, if necessary, to isolate 'a' on one side of the equation.
Solving for Acceleration in Each Equation Let's analyze each equation:
t = a Δ v . To solve for 'a', we can multiply both sides by 'a' and then divide by 't': a = t Δ v . This equation is a valid expression for acceleration.
v f = a t − v i . To solve for 'a', we first add v i to both sides: v f + v i = a t . Then, we divide by 't': a = t v f + v i . This equation is also a valid expression for acceleration.
a = t d . This equation is already solved for 'a'. 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 finding acceleration.
Δ v = t a . To solve for 'a', we multiply both sides by 't': a = t Δ v . This equation is not a valid expression for acceleration. Acceleration is defined as the rate of change of velocity, so a = t Δ v .
Identifying Correct Equations From the analysis above, equations 1 and 2 can be rearranged to solve for acceleration. However, equation 3 is not a general definition of acceleration, and equation 4 is incorrect.
Examples
Understanding acceleration is crucial in many real-world scenarios. For instance, when designing a car, engineers need to calculate the acceleration required to reach a certain speed within a specific time frame. Similarly, in sports, analyzing a sprinter's acceleration helps coaches optimize training techniques. These calculations rely on the fundamental relationships between acceleration, velocity, and time, as expressed in the equations we've explored.
Equations 1 ( t = a Δ v ) and 2 ( v f = a t − v i ) can be rearranged to solve for acceleration. Equation 3 does not correctly pertain to acceleration, and Equation 4 is incorrect. Thus, the valid equations for finding acceleration are Equations 1 and 2.
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