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Question

When we used derivation of motion in numericals

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Solution

The three equations of motions are:

1)*v* = v0 + *a*Δ*t*

*2) x* = x0 + v0Δ*t* + ½*a*Δ*t^2*

3)*v^2* = v0*^2* + 2*a*(*x* − *x*0)

These are the equations you learn in the physics class at school to calculate the velocity or end position of an object given an acceleration and a time or a distance.

Let me tell you a little secret: there are no 3 equations of motion. The three things you see above follow directly from the definition of a constant acceleration in one direction. But you need a bit of calculus to see that. The 3 above are an easy way to calculate things without needing calculus.

To know which one you need requires a close examination of your physics problem. E.g.

Lets try this out!

**Equation 1**: gives you a speed as answer and it needs a time as input.

Answer:

v0 = 10 m/s ; a = 0 m/s^2 (she isn't accelerating) ; Δ*t *= 10 s

So, v = 10 + 0 * 10 = 10 m/s

Answer:

v0 = 10 m/s ; a = -10 m/s^2 ; Δ*t *= 2 s

So, v = 10 + -10 * 2 = - 10 m/s (she travels in the opposite direction)

**Equation 2: **gives you a position as answer and needs a time as input.

Answer:

x0 = A (some point in space) ; v0 = 10 m/s ; a = 0 ; Δ*t = 10 s*

So, x = A + 10 * 10 + ½ * 0 * 2^ 2 = A + 100 meter.

I don't know where A is but Jan is 100 meters further on her route.

Answer:

x0 = A (some point in space) ; v0 = 10 m/s ; a = -10 m/s^2 ; Δ*t = 2 s*

So, x = A + 10 * 2 + ½ * - 10 * 2 ^ 2 = A + 20 - 20 = A

I don't know where A is but that is where Jan is now.

**Equation 3: **gives you a speed given a distance.

Answer:

v0 = 10 ; a = -10 m/s^2 ; x - x0 = 0 (she moved right turned and is back in A).

v ^ 2 = 10 ^ 2 + 2 * -10 * 0 = 100 ; | v | = 10 m/s^2 We know that the magnitude of the speed is 10 m/s, but because equation 3 uses squares we don't know the direction.

1)

3)

These are the equations you learn in the physics class at school to calculate the velocity or end position of an object given an acceleration and a time or a distance.

Let me tell you a little secret: there are no 3 equations of motion. The three things you see above follow directly from the definition of a constant acceleration in one direction. But you need a bit of calculus to see that. The 3 above are an easy way to calculate things without needing calculus.

To know which one you need requires a close examination of your physics problem. E.g.

- Equation 1) you use when the physics teacher asks you to calculate a
**speed**, given some actions during a period of**time**. - Equation 2) you use when the physics teacher asks you to calculate a
**position**, given some actions during a period of**time**. - Equation 3) you use when the physics teachers asks you to calculate a
**speed**, given some actions over some**distance**.

Lets try this out!

- What is the
**speed**of Jan when she travels at 10 m/s without accelerating for 10 seconds?

Answer:

v0 = 10 m/s ; a = 0 m/s^2 (she isn't accelerating) ; Δ

So, v = 10 + 0 * 10 = 10 m/s

- What is the
**speed**of Jan when she travels at 10/ms and decelerates at 10 m/s^2 for 2 seconds?

Answer:

v0 = 10 m/s ; a = -10 m/s^2 ; Δ

So, v = 10 + -10 * 2 = - 10 m/s (she travels in the opposite direction)

- Jan travels at 10 m/s without acceleration
**where**is she 10 seconds later?

Answer:

x0 = A (some point in space) ; v0 = 10 m/s ; a = 0 ; Δ

So, x = A + 10 * 10 + ½ * 0 * 2^ 2 = A + 100 meter.

I don't know where A is but Jan is 100 meters further on her route.

- Jan travels at 10 m/s , decelerates at 10 m/s^s for 2 seconds,
**where**is she now?

Answer:

x0 = A (some point in space) ; v0 = 10 m/s ; a = -10 m/s^2 ; Δ

So, x = A + 10 * 2 + ½ * - 10 * 2 ^ 2 = A + 20 - 20 = A

I don't know where A is but that is where Jan is now.

- Jan travels at 10 m/s , from the last problem we know she decelerated at 10 m/s^2 and is again at A. What is her speed?

Answer:

v0 = 10 ; a = -10 m/s^2 ; x - x0 = 0 (she moved right turned and is back in A).

v ^ 2 = 10 ^ 2 + 2 * -10 * 0 = 100 ; | v | = 10 m/s^2 We know that the magnitude of the speed is 10 m/s, but because equation 3 uses squares we don't know the direction.

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