Acceleration, Projectiles & Kinematics

Velocity is the rate of change of an object's position, and therefore has direction. This makes it a vector quantity, unlike speed which is scalar (magnitude but no direction).

velocity = distance / time

The units of velocity are m/s

Acceleration is the rate of change of velocity, and so is the mathematical derivative of this.

acceleration = change in velocity / change in time

The units of acceleration are m/s²

Motion Graphs

We can plot motion on two main types of graph - it is important to know the properties of each.

Displacement-Time Graphs

• The Gradient is the velocity - draw a tangent to find the instantaneous velocity

• Horizontal line represents zero velocity

Velocity-Time Graphs

• The Gradient is the acceleration

• The Area beneath the graph is the displacement

Constant Acceleration

When acceleration is constant (e.g. free fall when we ignore air resistance), we can use SUVAT equations to work out the variables:

• s is for displacement

• u is for initial velocity

• v is for final velocity

• a is for acceleration

• t is for time

Vertical Motion due to Gravity

The gravitational force of the earth causes all objects to accelerate towards the ground, its surface. Ignoring air resistance, the acceleration is constant, and given a g:

g = 9.8 m/s²

This is independent of the mass, shape or velocity of the object.

It is vital to set a positive direction of motion for each question

In the example above, we set up as the positive direction, so our values for a and v were negative because the ball is going down.

Constant Acceleration with Vectors

Additionally, we can express motion using vectors:

r = r₀ + v t

• r is the position vector of the moving object

• r₀ is the initial position vector

• v is the velocity vector

Four of the five SUVAT equations have vector equivalents:

• v = u + at v = u + a t

• s = ut + ½at² s = u t + ½ a t² + r₀

• s = vt - ½at² s = v t - ½ a t² + r₀

• s = ½(u+v)t s = ½( u + v )t + r₀

v² = u² + 2as has no vector equivalent

Projectile Motion

When we model a projectile, we ignore air resistance. This means that:

Horizontal motion of a projectile has constant velocity: a = 0

Vertical motion of a projectile motion is modelled as gravitational free fall: a = g

For horizontal projection, like in the diagram above:

• Since horizontal velocity is constant, we can use the equation x = vt (were x is horizontal displacement)

• For vertical velocity, we need to use SUVAT equations, due to the constant acceleration of g = 9.8 m/s²

Horizontal and Vertical Components

When given the velocity as a vector or at an angle, you must use trigonometry to find the vertical and horizontal components. Then, treat them separately as above (the horizontal component still has constant velocity)