**Kinematic equations** are the set of 4 equations that consist of five kinematic variables: **displacement** (Δx), **acceleration** (a), **initial velocity** (v_{i}), **final velocity** (v_{f}) 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:

- When “Δx” is missing, use the equation:
**v**_{f}**= v**_{i}**+ a t**

- When “a” is missing, use the equation:
**Δx = ½ × (v**_{i}**+ v**_{f}**) t**

- When “v
_{f}” is missing, use the equation:**Δx = v**_{i}**t + ½ a t**^{2}

- When “t” is missing, use the equation:
**v**_{f}^{2}**= v**_{i}^{2}**+ 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, v_{i} = 0 m/s

Final velocity of a bicycle, v_{f} = 75 m/s

Time taken by a bicycle, t = 15 s

Acceleration of a bicycle, a = ?

Using the kinematic equation, (when “Δx” is missing)

v_{f} = v_{i} + a t

75 = 0 + (a × 15)

75 = a × 15

a = 75/15

a = 5 m/s^{2}

Therefore, the acceleration of a bicycle is **5 m/s ^{2}**.

**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, v_{i} = 4 m/s

Final velocity of an athlete, v_{f} = 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 × (v_{i} + v_{f}) 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/s^{2} for 4 s. Calculate the displacement of a car during this time interval.

Solution:

Given data:

Initial velocity of a car, v_{i} = 0 m/s

Acceleration of a car, a = 15 m/s^{2}

Time taken by a car, t = 4 s

Displacement of a car, Δx = ?

Using the kinematic equation, (when “v_{f}” is missing)

Δx = v_{i} t + ½ a t^{2}

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/s^{2}. Calculate the displacement of a bike.

Solution:

Given data:

Initial velocity of a bike, v_{i} = 20 m/s

Final velocity of a bike, v_{f} = 0 m/s

Deceleration of a bike, a = -8 m/s^{2}

Displacement of a bike, Δx = ?

Using the kinematic equation, (when “t” is missing)

v_{f}^{2} = v_{i}^{2} + 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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