# 3.11.4 Differentiation - Kinematics

#### What is kinematics?

• Kinematics is the analysis of the motion of a particle linking the three vector quantities displacement, velocity and acceleration – see below
• Motion is in a straight line – think of the particle as moving along a number line
• The number line has a fixed point O (the origin)
• The number line has both negative and positive values
• The particle can move in both directions along the number line
• Note that in kinematics, a particle is an object – it could be a football, a car, a train – anything that has motion.  A particle is modelled as taking up a single point in space • Ensure you are familiar with Differentiation – Basics and
Differentiation – Turning Points before continuing
• It may be wise to look at Differentiation – Problem Solving too

#### What is displacement; isn’t it the same as distance?

• Displacement is a vector quantity, so it can be negative
• Distance is always positive
• Displacement is measured from the fixed point O
• The letter s is used for displacement
• It is usually measured in metres (m)
• If s = 4 then the distance from the origin is 4 m and the particle is 4 m “in front of” the origin
• If s =-5 then the distance from the origin is 5 m and the particle is 5 m “behind” the origin
• The + or – indicates the particle’s position relative to the origin • Displacement is a function of time, t, where time is usually measured in seconds
• eg. s = 3t3 – 2t + 1
At time t = 0, s = 1
At time t = 2, s = 21 #### What is velocity; isn’t it the same as speed?

• Velocity is a vector quantity, so it can be negative
• Speed is always positive
• The letter v is used for velocity
• It is usually measured in metres per second (m/s)
• If v = 3 then the speed of the particle is 3 m/s and it is moving in the positive direction
• If v = -6 then the speed of the particle is 5 m/s and it is moving in the negative direction
• The + or – indicates the particle’s direction of motion • Velocity is a function of time, t, and is the rate of change of displacement
• To find v, differentiate s, ie. v = ds/dt
If s = t3 – 2t2
then v = ds/dt = 3t2 – 4t
• If velocity is zero then the particle is stationary (not moving) #### What is acceleration?

• Acceleration is a vector quantity, so it can be negative
• The magnitude of acceleration is always positive
• The letter a is used for acceleration
• It is usually measured in metres per square second (m/s2)
• If a = 1 then the magnitude of acceleration is 1 m/s2 and the particle is accelerating (velocity increasing)
• If a = -6 then the magnitude of acceleration is 6 m/s2 and the particle is decelerating (velocity decreasing)
• The + or – indicates whether the particle is accelerating or decelerating • Acceleration is a function of time, t, and is the rate of change of velocity
• To find a, differentiate v, ie. a = dv/dt
If v = 3t2 – 4t
then a = dv/dt = 6t – 4
• If acceleration is zero then the particle is moving at a constant velocity #### How do I solve kinematics problems?

• Be clear about how the three quantities are related through differentiation
• v = ds/dt
• a = dv/dt • There are some key phrases to look out for
• “… initial …” / “… initially …”
This means at the start, so when t = 0
• “… at rest …”
This means the particle is stationary
so v = 0
• “… instantaneously …”
This means at some point in time, for some value of t
• For example,
“Find the value(s)s of t for which the particle is instantaneously at rest”

• means find the time(s) when v = 0,
• ie. solve the equation v = 0

#### Exam Tip

Displacement, velocity and acceleration can all be negative whereas distance, speed and magnitude of acceleration are always positive.

### Worked Example

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