Kinematics Equations

Motion

What is kinematics?

The study and description of the motion of objects.

What is uniform motion?

Motion with no acceleration, so velocity remains constant.

What is the equation for velocity during uniform motion?

v = s/t.

What is the equation for displacement when velocity is constant?

s = vt.

What does s represent in kinematics equations?

Displacement, measured in metres (m).

What does u represent in kinematics equations?

Initial velocity, measured in m s⁻¹.

What does v represent in kinematics equations?

Final velocity, measured in m s⁻¹.

What does a represent in kinematics equations?

Acceleration, measured in m s⁻².

What does t represent in kinematics equations?

Time, measured in seconds (s).

When can the constant-acceleration kinematics equations be used?

Only when acceleration is constant.

What should you do first when solving a kinematics problem?

Identify the known and unknown values of s, u, v, a and t.

What does it mean if an object starts from rest?

Its initial velocity is zero, so u = 0 m s⁻¹.

What acceleration is normally used for an object falling freely near Earth's surface?

g = 9.81 m s⁻².

What is the first constant-acceleration equation?

v = u + at.

What does v = u + at calculate?

Final velocity when initial velocity, acceleration and time are known.

How can v = u + at be rearranged to find acceleration?

a = (v − u)/t.

How can v = u + at be rearranged to find initial velocity?

u = v − at.

How can v = u + at be rearranged to find time?

t = (v − u)/a.

What is the equation linking displacement, initial velocity, final velocity and time?

s = ((u + v)/2)t.

Why is (u + v)/2 used in s = ((u + v)/2)t?

For constant acceleration, it gives the average velocity.

How do you calculate average velocity during constant acceleration?

Average velocity = (u + v)/2.

How is displacement calculated from average velocity?

Displacement = average velocity × time.

How can s = ((u + v)/2)t be rearranged to find time?

t = 2s/(u + v).

What is the equation for displacement when final velocity is not known?

s = ut + ½at².

What does s = ut + ½at² calculate?

Displacement using initial velocity, acceleration and time.

How can s = ut + ½at² be simplified for an object starting from rest?

s = ½at².

How can s = ut + ½at² be simplified for an object dropped from rest under gravity?

s = ½gt².

What is the kinematics equation that does not contain time?

v² = u² + 2as.

What does v² = u² + 2as calculate?

It links initial velocity, final velocity, acceleration and displacement without using time.

How can v² = u² + 2as be rearranged to find acceleration?

a = (v² − u²)/(2s).

How can v² = u² + 2as be rearranged to find displacement?

s = (v² − u²)/(2a).

How can v² = u² + 2as be rearranged to find final velocity?

v = √(u² + 2as).

Which kinematics equation does not use displacement?

v = u + at.

Which kinematics equation does not use acceleration?

s = ((u + v)/2)t.

Which kinematics equation does not use final velocity?

s = ut + ½at².

Which kinematics equation does not use time?

v² = u² + 2as.

How do you choose the correct kinematics equation?

Identify the three known quantities and the unknown, then choose the equation containing those four quantities.

What are the four main constant-acceleration equations?

v = u + at; s = ((u + v)/2)t; s = ut + ½at²; v² = u² + 2as.

Why must acceleration be constant when using the kinematics equations?

The equations are derived assuming uniform, constant acceleration.

Can the kinematics equations be used if acceleration changes?

No, not directly, because they assume constant acceleration.

What is free fall?

Motion where gravity is the only significant force acting on an object.

What acceleration does an object in free fall experience near Earth's surface?

Approximately 9.81 m s⁻² downwards.

What value of u is used when an object is dropped rather than thrown?

u = 0 m s⁻¹.

What sign should gravitational acceleration have?

It depends on the chosen positive direction; if upwards is positive, a = −9.81 m s⁻², while if downwards is positive, a = +9.81 m s⁻².

Why is choosing a positive direction important in kinematics?

It determines the signs of displacement, velocity and acceleration.

A stone is dropped from rest for 3 s. What is its speed after 3 s, ignoring air resistance?

v = 0 + 9.81 × 3 = 29.4 m s⁻¹.

A stone falls from rest for 3 s. How far does it fall?

s = ½ × 9.81 × 3² = 44.1 m.

What should happen if two different valid kinematics equations are used for the same motion?

They should give the same answer.

How can the acceleration due to gravity g be measured experimentally?

Drop an object from different known heights and measure its fall time.

For an object dropped from rest, what equation links height and fall time?

h = ½gt².

How can h = ½gt² be rearranged?

t² = 2h/g.

What graph can be plotted to determine g from free-fall measurements?

Plot t² on the y-axis against h on the x-axis.

What is the gradient of a graph of t² against h?

2/g.

How can g be calculated from the gradient of a t² against h graph?

g = 2/gradient.

Why are several measurements used in a free-fall experiment instead of one?

Multiple measurements and a graph reduce the effect of random uncertainty and give a more reliable value of g.

What is a random error?

An unpredictable variation between repeated measurements.

What is a systematic error?

An error that consistently shifts measurements in the same direction.

Why is plotting a graph useful in an experiment?

It allows a best-fit relationship to be found using many measurements and reduces the influence of individual random errors.

What safety precaution should be taken when using a tall stand for a free-fall experiment?

Secure the stand so that it cannot topple over.

A bike starts at 4 m s⁻¹ and accelerates at 1 m s⁻² for 10 s. What is its final velocity?

v = 4 + (1 × 10) = 14 m s⁻¹.

A bike changes velocity from 4 m s⁻¹ to 14 m s⁻¹ in 10 s. What is its average velocity?

(4 + 14)/2 = 9 m s⁻¹.

If that bike has an average velocity of 9 m s⁻¹ for 10 s, how far does it travel?

s = 9 × 10 = 90 m.

A ball starts from rest and reaches 4.8 m s⁻¹ after travelling 30 m. Which equation should be used to find its acceleration?

v² = u² + 2as because time is not given.

A train travels 100 m from rest in 15 s with constant acceleration. Which equation should be used to find its acceleration?

s = ut + ½at² because s, u and t are known.

What is the difference between speed and velocity in kinematics?

Speed is a scalar, while velocity includes both magnitude and direction.

What is the difference between distance and displacement?

Distance is the total path travelled, while displacement is the change in position in a specified direction.