# Kinematic Equations | Problems (With Solutions)

Kinematic equations are the set of 4 equations that consist of five kinematic variables: displacement (Δx), acceleration (a), initial velocity (vi), final velocity (vf) and time (t).

Each kinematic equation contains four known kinematic variables and one unknown kinematic variable. Using the proper kinematic equation, the unknown information about an object’s motion can be found out.

(Note that the kinematic equations are used only for an object undergoing constant acceleration. These equations are not used if the acceleration of an object is changing)

Here are the 4 different kinematic equations:

1. When “Δx” is missing, use the equation: vf = vi + a t
1. When “a” is missing, use the equation: Δx = ½ × (vi + vf) t
1. When “vf” is missing, use the equation: Δx = vi t + ½ a t2
1. When “t” is missing, use the equation: vf2 = vi2 + 2 a Δx

Let’s solve some problems based on these equations, so you’ll get a clear idea.

## Kinematic Equations Practice Problems

Problem 1: Calculate the acceleration of a bicycle which accelerates from 0 m/s to 75 m/s in 15 s.

Solution:

Given data:
Initial velocity of a bicycle, vi = 0 m/s
Final velocity of a bicycle, vf = 75 m/s
Time taken by a bicycle, t = 15 s
Acceleration of a bicycle, a = ?

Using the kinematic equation, (when “Δx” is missing)
vf = vi + a t
75 = 0 + (a × 15)
75 = a × 15
a = 75/15
a = 5 m/s2

Therefore, the acceleration of a bicycle is 5 m/s2.

Problem 2: An athlete running in the olympic race speeds up from 4 m/s to 12 m/s in 6 s. Calculate the displacement of an athlete.

Solution:

Given data:
Initial velocity of an athlete, vi = 4 m/s
Final velocity of an athlete, vf = 12 m/s
Time taken by an athlete, t = 6 s
Displacement of an athlete, Δx = ?

Using the kinematic equation, (when “a” is missing)
Δx = ½ × (vi + vf) t
Δx = ½ × [(4 +12 ) × 6]
Δx = 16 × 3
Δx = 48 m

Therefore, the displacement of an athlete is 48 m.

Problem 3: A car starts from rest and accelerates at the rate of 15 m/s2 for 4 s. Calculate the displacement of a car during this time interval.

Solution:

Given data:
Initial velocity of a car, vi = 0 m/s
Acceleration of a car, a = 15 m/s2
Time taken by a car, t = 4 s
Displacement of a car, Δx = ?

Using the kinematic equation, (when “vf” is missing)
Δx = vi t + ½ a t2
Δx = [0 × (4)] + [½ × 15 × (4)2]
Δx = ½ × 15 × 16
Δx = 120 m

Therefore, the displacement of a car is 120 m.

Problem 4: One boy is riding a bike on the highway with a velocity of 20 m/s. In order to stop a bike at the gas station, he applies the brakes to a bike. So a bike slows down with a deceleration of 8 m/s2. Calculate the displacement of a bike.

Solution:

Given data:
Initial velocity of a bike, vi = 20 m/s
Final velocity of a bike, vf = 0 m/s
Deceleration of a bike, a = -8 m/s2
Displacement of a bike, Δx = ?

Using the kinematic equation, (when “t” is missing)
vf2 = vi2 + 2 a Δx
(0)2 = (20)2 + [2 × (-8) × Δx]
16 × Δx = (20)2
16 × Δx = 400
Δx = 400/16
Δx = 25 m

Therefore, the displacement of a bike is 25 m.

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