**Kinematic Equations Formula**

#### Kinematics is the study of objects in motion and their inter-relationships. There are four (4) kinematic equations, which relate to displacement, D, velocity, v, time, t, and acceleration, a.

**a) ***D = v*_{i}t + 1/2 at^{2} b)* (v*_{i} +v_{f})/2 = D/t

*D = v*b)

_{i}t + 1/2 at^{2}*(v*

_{i}+v_{f})/2 = D/t**c)*** a = (v*_{f} – v_{i})/t d) *v*_{f}^{2} = v_{i}^{2} + 2aD

*a = (v*d)

_{f}– v_{i})/t*v*

_{f}^{2}= v_{i}^{2}+ 2aD*D* = displacement

*a* = acceleration

*t* = time

*v*_{f} = final velocity

_{f}

*v*_{i} = initial velocity

_{i}

####
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**Kinematic Equations Formula Questions.**

#### 1) Bob is riding his bicycle to the store at a velocity of 4 m/s, when a cat runs out in front of him. He quickly brakes to a complete stop, with an acceleration of – 2m/s^{2}. What is his displacement?

**Answer:** Because Bob is stopped, the final velocity, v_{f} = 0. His initial velocity, v_{i} = 4 m/s. The acceleration, a = -2m/s^{2}. Time is not given, so use equation (d) for displacement, D, because it is not time-dependent.

*v*_{f}^{2} = v_{i}^{2} + 2aD

*v*

_{f}^{2}= v_{i}^{2}+ 2aD#### (0)^{2}= (4 m/s)^{2} +2(- 2 m/s^{2})D

#### 0 = 16 m^{2}/s^{2} + (- 4m/s^{2})D

#### -16 m^{2}/s^{2} = (- 4 m/s^{2})D

#### 16 m^{2}/s^{2} = 4 m/s^{2})D

#### (16 m^{2}/s^{2}) / (4 m/s^{2}) = D

#### The total displacement is 4 m.

####
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#### 2) You travel at a constant velocity of 11 m/s for 5 minutes. How far have you traveled?

**Answer:** At constant velocity, v_{i} = v_{f} = 11 m/s. The time, t = 5 min, or t = (60 sec/min x 5 min) = 300 sec. Now use equation (b) to solve for displacement, D.

*(v*_{i} + v_{f})/2 = D/t

*(v*

_{i}+ v_{f})/2 = D/t*D = [(v*_{i} + v_{f})/2] t

_{i}+ v

_{f})/2] t

*D *= [(11 m/s + 11 m/s)/2] x 300 sec

*D* = (22 m/s)/2 x 300 sec

*D* = 11 m/s x 300 sec

*D* = 3,300 m The total displacement is 3, 300 m.

####
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#### 3) What is the acceleration of a car that speeds up from 11 m/s to 40 m/s after 10 seconds?

**Answer:** The V_{i} = 11 m/s. The v_{f} = 40 m/s. Time, t = 10 s. Use kinematic equation c) to solve for acceleration.

*a = (v*_{f} – v_{i})/t

*a = (v*

_{f}– v_{i})/t*a* = (40 m/s – 11 m/s) /10 s

*a* = (29m/s)/10 s = 2.9 m/s^{2}

#### 4) If a car accelerates at 3.0 m/s^{2} from a complete stop, how long will it take to go 3000 m?

**Answer:** The acceleration, a = 2.9 m/s^{2}, and the displacement, D = 3000 m. The car was at rest, so v_{i} = 0. Use equation a) to solve for time.

*D = v*_{i}t + 1/2 at^{2}

_{i}t + 1/2 at

^{2}

*3000 m = 0t + 1/2 (3.0 m/s*^{2})t^{2}

^{2})t

^{2}

*3000 m = 1/2 (3.0 m/s*^{2})/t^{2}

^{2})/t

^{2}

*3000 m/ 1.5 m/s*^{2} = t^{2}

^{2}= t

^{2}

*2000 s*^{2} = t^{2}

^{2}= t

^{2}