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Sagot :
To prove the given equation [tex]\( s = u t + \frac{7}{2} a t^2 \)[/tex], we will break down the expression and show each component's derivation and combination step by step.
### Step 1: Understand the equation
The equation represents a second-degree polynomial that relates displacement [tex]\( s \)[/tex] to time [tex]\( t \)[/tex], initial velocity [tex]\( u \)[/tex], and acceleration [tex]\( a \)[/tex].
### Step 2: Identify the terms
1. Displacement term: [tex]\( s \)[/tex]
2. Initial velocity term: [tex]\( u t \)[/tex]
3. Acceleration term: [tex]\( \frac{7}{2} a t^2 \)[/tex]
### Step 3: Explanation
#### Term 1: Initial velocity contribution
The term [tex]\( u t \)[/tex] represents the distance traveled due to the initial velocity [tex]\( u \)[/tex] over time [tex]\( t \)[/tex]. This is a straightforward application of the basic kinematic equation where:
[tex]\[ \text{Displacement due to initial velocity} = \text{Initial velocity} \times \text{Time} \][/tex]
#### Term 2: Acceleration contribution
The term [tex]\( \frac{7}{2} a t^2 \)[/tex] represents the additional distance traveled due to the acceleration [tex]\( a \)[/tex] over time [tex]\( t \)[/tex]. This term is obtained by considering that acceleration causes the object to cover more distance compared to its initial velocity alone.
In general kinematics, under constant acceleration, the distance traveled due to acceleration is usually given as [tex]\( \frac{1}{2} a t^2 \)[/tex]. Here, rather than [tex]\( \frac{1}{2} \)[/tex], we have [tex]\( \frac{7}{2} \)[/tex], indicating a different factor influencing this acceleration term, possibly due to varying specific conditions or a different context which is not detailed here.
### Step 4: Combine the terms
By summing the contributions from both initial velocity and acceleration, we get the total displacement [tex]\( s \)[/tex] as a function of [tex]\( t \)[/tex], [tex]\( u \)[/tex], and [tex]\( a \)[/tex]:
[tex]\[ s = u t + \frac{7}{2} a t^2 \][/tex]
Combining these individual components, we arrive at the final equation:
[tex]\[ s = u t + \frac{7}{2} a t^2 \][/tex]
This demonstrates how the object's motion can be described through both its initial velocity and the effect of constant acceleration.
### Conclusion
Thus, the equation is proved and each term’s significance in terms of physical kinematics is thoroughly explained. The final form:
[tex]\[ s = u t + \frac{7}{2} a t^2 \][/tex]
accurately represents the displacement [tex]\( s \)[/tex] at time [tex]\( t \)[/tex], considering both initial velocity [tex]\( u \)[/tex] and acceleration [tex]\( a \)[/tex].
### Step 1: Understand the equation
The equation represents a second-degree polynomial that relates displacement [tex]\( s \)[/tex] to time [tex]\( t \)[/tex], initial velocity [tex]\( u \)[/tex], and acceleration [tex]\( a \)[/tex].
### Step 2: Identify the terms
1. Displacement term: [tex]\( s \)[/tex]
2. Initial velocity term: [tex]\( u t \)[/tex]
3. Acceleration term: [tex]\( \frac{7}{2} a t^2 \)[/tex]
### Step 3: Explanation
#### Term 1: Initial velocity contribution
The term [tex]\( u t \)[/tex] represents the distance traveled due to the initial velocity [tex]\( u \)[/tex] over time [tex]\( t \)[/tex]. This is a straightforward application of the basic kinematic equation where:
[tex]\[ \text{Displacement due to initial velocity} = \text{Initial velocity} \times \text{Time} \][/tex]
#### Term 2: Acceleration contribution
The term [tex]\( \frac{7}{2} a t^2 \)[/tex] represents the additional distance traveled due to the acceleration [tex]\( a \)[/tex] over time [tex]\( t \)[/tex]. This term is obtained by considering that acceleration causes the object to cover more distance compared to its initial velocity alone.
In general kinematics, under constant acceleration, the distance traveled due to acceleration is usually given as [tex]\( \frac{1}{2} a t^2 \)[/tex]. Here, rather than [tex]\( \frac{1}{2} \)[/tex], we have [tex]\( \frac{7}{2} \)[/tex], indicating a different factor influencing this acceleration term, possibly due to varying specific conditions or a different context which is not detailed here.
### Step 4: Combine the terms
By summing the contributions from both initial velocity and acceleration, we get the total displacement [tex]\( s \)[/tex] as a function of [tex]\( t \)[/tex], [tex]\( u \)[/tex], and [tex]\( a \)[/tex]:
[tex]\[ s = u t + \frac{7}{2} a t^2 \][/tex]
Combining these individual components, we arrive at the final equation:
[tex]\[ s = u t + \frac{7}{2} a t^2 \][/tex]
This demonstrates how the object's motion can be described through both its initial velocity and the effect of constant acceleration.
### Conclusion
Thus, the equation is proved and each term’s significance in terms of physical kinematics is thoroughly explained. The final form:
[tex]\[ s = u t + \frac{7}{2} a t^2 \][/tex]
accurately represents the displacement [tex]\( s \)[/tex] at time [tex]\( t \)[/tex], considering both initial velocity [tex]\( u \)[/tex] and acceleration [tex]\( a \)[/tex].
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