# Kinematic equations

The kinematic equations refer to a set of four equations that involve five key variables: displacement (Δx), acceleration (a), initial velocity (vi), final velocity (vf), and time (t). These equations serve as a powerful tool for analyzing the motion of objects. Each equation contains four known kinematic variables and one unknown variable, allowing us to determine missing information about an object’s motion. It’s important to note that the kinematic equations are specifically applicable when an object undergoes constant acceleration, and they are not used when the acceleration is changing.

The four different kinematic equations are:

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

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## Practice problems

### Problem #1

Calculate the acceleration of a bicycle that goes from 0 m/s to 75 m/s in 15 s.

Solution

Given data:

• Acceleration of a bicycle, a = ?
• 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

Applying the formula, 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 participating in the Olympic race accelerates from 4 m/s to 12 m/s in 6 s. Calculate the displacement of the 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 = ?

Applying the formula, 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 a rate of 15 m/s2 for 4 s. Calculate the displacement of the 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 = ?

Applying the formula, 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

A boy is riding a bike on the highway with a velocity of 20 m/s. To stop the bike at the gas station, he applies the brakes, causing the bike to decelerate at 8 m/s2. Calculate the displacement of the 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 = ?

Applying the formula, 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.