Variable Acceleration in A Level Mechanics
Variable Acceleration – Introduction
Introduction:
In A Level Mechanics, the concept of variable acceleration is studied to understand the motion of objects where acceleration is not constant. This topic explores how the rate of change of velocity varies with time, leading to a more complex analysis of motion. Calculus plays a crucial role in working with variable acceleration, enabling us to derive equations and solve problems related to this topic.
Definition of Variable Acceleration:
Variable acceleration refers to the situation where an object’s acceleration changes over time. Unlike constant acceleration, where the rate of change of velocity remains the same, variable acceleration involves a changing rate of velocity. This change can be either positive (acceleration increasing) or negative (acceleration decreasing) or a combination of both.
Equations of Motion with Variable Acceleration:
When dealing with variable acceleration, we need to modify the equations of motion accordingly. The following equations are commonly used:
Displacement as a function of time:
s(t) = s₀ + v₀t + ½at²
Velocity as a function of time:
v(t) = v₀ + at
Velocity as a function of displacement:
v² = v₀² + 2a(s – s₀)
Displacement as a function of velocity:
s(t) = s₀ + ∫[v₀ + at] dt
The Need for Calculus:
Calculus is essential in working with variable acceleration as it allows us to calculate quantities such as instantaneous velocity, displacement, and acceleration. By taking derivatives and integrals, we can find the rate of change, the area under a curve, and solve complex problems involving changing rates of motion.
Derivatives in Variable Acceleration:
To determine instantaneous velocity and acceleration at a given point, we use derivatives. By taking the derivative of the displacement equation, we can obtain the velocity equation:
v(t) = ds/dt = v₀ + at
Similarly, taking the derivative of the velocity equation gives the acceleration equation:
a(t) = dv/dt = a
Integrals in Variable Acceleration:
Integrals are employed to find displacement when velocity is not constant. By integrating the velocity equation, we can calculate the displacement equation:
s(t) = ∫[v₀ + at] dt = s₀ + v₀t + ½at²
Integration is also used to analyse the area under velocity-time graphs. The area under the curve represents the displacement, and integrating the acceleration equation helps determine the change in velocity.
Solving Problems with Variable Acceleration:
Using the equations mentioned earlier and applying calculus techniques, we can solve a variety of problems related to variable acceleration. These may include calculating time of flight, maximum height, distance travelled, or determining the shape of a trajectory.
Variable Acceleration - Examples
More Variable Acceleration - Examples
3.
4.
Part a)
\begin{aligned} & v=12-2 t^2 \\ & s=12 t-\frac{2 t^3}{3}+c \end{aligned}\begin{aligned} & s=0 \quad t=0 \\ & s=12 t-\frac{2}{3} t^3 \\ & s=12(1)-\frac{2}{3}(1)^3 \\ &=\frac{34}{3} \mathrm{~m} \end{aligned}
Part b)
\begin{aligned} 0 & =12-2 t^2 \\ 2 t^2 & =12 \\ t^2 & =6 \\ t & =\sqrt{6} \mathrm{~s} \end{aligned}
Part c)
\begin{aligned} & 0=12 t-\frac{2}{3} t^3 \\ & 0=t\left(12-\frac{2}{3} t^2\right) \\ & t=0 \text { or } 12-\frac{2}{3} t^2=0 \\ & 12=\frac{2}{3} t^2 \\ & 18=t^2 \\ & t=\sqrt{18} \\ & =3 \sqrt{2} \mathrm{~s} \\ & \end{aligned}
Variable Acceleration – Final Example
Part a)
\begin{aligned} & x= 2 t^3-18 t^2+48 t \\ & v= 6 t^2-36 t+48 \\ & 6 t^2-36 t+48=0 \\ & t^2-6 t+8=0 \\ &(t-2)(t-4)=0 \end{aligned}
Part b)
t=0 \quad s=0
\begin{aligned} \epsilon=2 \quad s & =2(2)^3-18(2)^2+48(2) \\ & =40 \end{aligned}
\begin{aligned} t=4 \quad s & =2(4)^3-18(4)^2+48(4) \\ & =32 \end{aligned}
\begin{aligned} t=5 \quad s & =2(5)^3-18(5)^2+48(5) \\ & =40 \end{aligned}
Total distance:
\begin{aligned} & =40+8+8 \\ & =56 \mathrm{~m} \end{aligned}Conclusion:
The concept of variable acceleration in A Level Mechanics allows for a more comprehensive understanding of object motion when acceleration is not constant. By utilising calculus, we can derive equations and solve problems related to this topic, enabling a deeper analysis of changing rates of velocity and acceleration.
