# 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/tc)* a = (v _{f} - v_{i})/t * d)

*v*

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

*a* = acceleration

*t* = time

*v _{f}* = final velocity

*v _{i}* = initial velocity

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*

(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.

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 *

*D = [(v _{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.

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* = (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}*

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

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

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

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

*t = 44.72 sec *

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