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 = vit + 1/2 at2 b) (vi +vf)/2 = D/t
c) a = (vf – vi)/t d) vf2 = vi2 + 2aD
D = displacement
a = acceleration
t = time
vf = final velocity
vi = 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/s2. What is his displacement?
Answer: Because Bob is stopped, the final velocity, vf = 0. His initial velocity, vi = 4 m/s. The acceleration, a = -2m/s2. Time is not given, so use equation (d) for displacement, D, because it is not time-dependent.
vf2 = vi2 + 2aD
(0)2= (4 m/s)2 +2(- 2 m/s2)D
0 = 16 m2/s2 + (- 4m/s2)D
-16 m2/s2 = (- 4 m/s2)D
16 m2/s2 = 4 m/s2)D
(16 m2/s2) / (4 m/s2) = 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, vi = vf = 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.
(vi + vf)/2 = D/t
D = [(vi + vf)/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 Vi = 11 m/s. The vf = 40 m/s. Time, t = 10 s. Use kinematic equation c) to solve for acceleration.
a = (vf – vi)/t
a = (40 m/s – 11 m/s) /10 s
a = (29m/s)/10 s = 2.9 m/s2
4) If a car accelerates at 3.0 m/s2 from a complete stop, how long will it take to go 3000 m?
Answer: The acceleration, a = 2.9 m/s2, and the displacement, D = 3000 m. The car was at rest, so vi = 0. Use equation a) to solve for time.
D = vit + 1/2 at2
3000 m = 0t + 1/2 (3.0 m/s2)t2
3000 m = 1/2 (3.0 m/s2)/t2
3000 m/ 1.5 m/s2 = t2
2000 s2 = t2
t = 44.72 sec