Revision Note 3: Kinematics

The following three equations of kinematics are valid only when acceleration is constant:

1.      v(t) = v(0) + at

2.      x(t) = x(0) + v(0)t + ½ at²

3.      [v(t)]² = [v(0)]² + 2a[x(t) – x(0)]

Here:

·        Time = t

·        Acceleration = a

·        Velocity at time t = v(t)

·        Position at time t = x(t)

Motion in 2D
The x- and y-directions have separate kinematics equations that do not interact. Only time connects the two.  So, the position of the particle at time t is (x(t), y(t)). If the particle is stopped (e.g., by falling to the ground), then the time of flight is the time when it hits the ground – generally this comes from the vertical kinematics equations.

If one direction (generally horizontal = x-direction) doesn’t have any acceleration, then the distance travelled in that direction is easy to find (x = vx . t). The total distance travelled in that direction then is velocity times time of flight.

Problem Solving Tips:

Tip 3.1:

When the acceleration is not constant it is expressed as a(t). Then use Calculus to find v(t), x(t) etc by integrating a(t).

Tip 3.2:

The time for a falling body to drop a height, h, is given by h = ½ gt².

Tip 3.3:

If you throw a particle vertically upwards with velocity, v, then the maximum height it reaches is given by the equation: v² = 2gh. This equation can be derived via kinematics and also from energy principles:

PE_i    +      KE_i       =    PE_f     +   KE_f

0       +   ½ mv²   =    mgh   +    0

        v²   =  2gh

Tip 3.4: What goes up …

What goes up in time, t, also comes down in time, t. In other words the equation from Tip 3.2 works in both directions – going up and going down. Not just that, the velocities going up and going down at any point in the flight have the same magnitude (the directions are opposites of each other).

Tip 3.5: Inclined planes

A particle pushed up an inclined plane (without friction) gains the same vertical height as a particle thrown vertically up. This follows from the energy principles as in Tip 3.3.

Even though the vertical heights are the same, the particle on the inclined plane travels a s longer distance and takes more time.

Tip 3.6: Up-down intuition:

Relating the height, h, from which a ball is dropped with the time, t, of flight:

h ↑ t ↑

The time goes up is less since for the latter part of the journey the speed of the ball is very high.

Relating the velocity, v, of a ball thrown up and the height, h:

v ↑ h ↑

Here too higher initial velocity cannot boost the height too much since the latter part of the journey is covered at lower speeds.

Tip 3.7: Projectile motion:

If v = initial velocity and θ = angle of launch (projectile launched from the ground level), then:

Maximum height = h = v².sin²θ / 2g

Range = R = v².sin 2θ / g

Time of flight = T = 2v.sin θ /g

The maximum value for h, T are achieved when θ=90 and for R when θ=45.

For maximum range: v_x = v_y at launch.