Solving Rectilinear Problems

 

The basic equations

 

Almost every particle rectilinear kinematic problem can be solved by manipulating the following three equations.

 

• Velocity: v = ds/dt
• Acceleration: a = dv/dt
• Acceleration as a function of position: a ds = v dv

 

Time-dependent equations

 

If the velocity or acceleration are dependent on time, the above velocity and acceleration equations may be integrated to find position or velocity. Note that when integrating, limits are required to obtain a solution.

 

• Velocity → Position: v = ds/dt   →   ∫v dt = ∫ds
• Acceleration → Velocity: a = dv/dt   →   ∫a dt = ∫dv

 

Position-dependent equations

 

If the velocity or acceleration are dependent on position, the following equation may be used.

 

• Acceleration depends on position: ∫a ds = ∫v dv
• Velocity depends on position: ∫dt = ∫(1/v)ds

 

Acceleration as a function of velocity

 

If the acceleration depends on velocity, the following equations may be used.

 

• Acceleration → Velocity: ∫dt = ∫(1/a) dv
• Acceleration → Position: ∫ds = ∫(v/a) dv

 

Constant-acceleration equations

 

If acceleration is constant, the following equations may be used.

 

• v = v₀ + a₀(t - t₀)
• v² = 2a₀(s - s₀) + v₀²
• s = s₀ + v₀(t - t₀) + a₀(t - t₀)²/2