Comprehensive assessment covering velocity and acceleration as derivatives, acceleration as a function of displacement, velocity or time, simple harmonic motion, projectile and resisted motion, and circular motion.
The velocity $v$ of a particle is defined as:
When acceleration is given as a function of displacement $x$, it is most usefully written as:
The expression $v\dfrac{dv}{dx}$ is equal to:
The defining equation of simple harmonic motion is:
A particle moves so that $x = 5\cos 3t$. Its amplitude and period are:
For simple harmonic motion $x = a\cos nt$, the maximum speed is:
For simple harmonic motion of amplitude $a$, the velocity satisfies $v^2$ equals:
For a projectile (ignoring air resistance), the horizontal and vertical accelerations are:
For a particle moving with simple harmonic motion, the acceleration is zero at:
A particle falls under gravity with resistance so that $\ddot{x} = g - kv$. Its terminal velocity is:
For uniform circular motion of radius $r$ and angular velocity $\omega$, the speed is:
In uniform circular motion, the acceleration is directed:
The magnitude of the centripetal acceleration in uniform circular motion (speed $v$, radius $r$) is:
If $x = a\sin(nt + \alpha)$, the period of the motion is:
If acceleration is given as a function of time $t$, the velocity is found by:
1. A particle has displacement $x = t^3 - 6t^2 + 9t$. Find its velocity and acceleration as functions of $t$. (2 marks)
2. A particle moves in simple harmonic motion with $x = 4\cos 2t$. State its amplitude, period and maximum speed. (2 marks)
3. For simple harmonic motion with amplitude $5$ and $n = 3$, find the maximum speed and the maximum acceleration. (2 marks)
4. A particle moving in simple harmonic motion satisfies $v^2 = 16 - 4x^2$. Find the amplitude and the value of $n$. (2 marks)
5. A particle starts from rest and moves with acceleration $a = 6t$. Given $x = 0$ when $t = 0$, find its velocity and displacement as functions of $t$. (3 marks)
6. Show that $x = a\cos(nt + \alpha)$ satisfies the equation $\ddot{x} = -n^2 x$. (2 marks)
7. Given $v\dfrac{dv}{dx} = -x$ and $v = 4$ when $x = 0$, find $v^2$ in terms of $x$. (2 marks)
8. A particle falling under gravity with resistance satisfies $\dfrac{dv}{dt} = 10 - 2v$. Find its terminal velocity. (1 mark)
9. A particle moves in a circle of radius $2$ m with constant angular velocity $3$ rad/s. Find its speed and the magnitude of its acceleration. (2 marks)
10. For the motion $x = 3\sin 2t$, find the times in $0 \le t \le \pi$ at which the particle is at the centre of motion. (2 marks)