Energy equation with constant resistance?

Initial energy - work done against resistance = final energy.

Does the normal reaction on a smooth surface do work?

No, it is perpendicular to the motion.

A particle slides from rest down a smooth slope of height h and stops on a rough surface with \mu: distance?

d=\frac h\mu; the mass cancels.

Does the speed of a projectile on landing depend on the angle?

No, only on the heights at start and end (ignoring air resistance).

When do you use energy instead of F=ma?

For speeds and distances; use F=ma for accelerations and forces.

Conservation of mechanical energyEdexcel International A Level Further Maths: Flashcards

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State the principle of conservation of mechanical energy.

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State the principle of conservation of mechanical energy.: If only conservative forces do work, KE plus PE is constant.
Speed after falling a height $h$ from rest?: $v=\sqrt{2gh}$
Energy equation with constant resistance?: Initial energy $-$ work done against resistance $=$ final energy.
Work done against a constant resistance $R$ over distance $d$ ?: $Rd$
Height gained moving $d$ up a slope at $\alpha$ ?: $d\sin\alpha$
Does the normal reaction on a smooth surface do work?: No, it is perpendicular to the motion.
Maximum height above ground of a vertical throw at $u$ from $h_0$ ?: $h_0+\frac{u^2}{2g}$
Distance to stop on a rough horizontal surface?: $d=\frac{v^2}{2\mu g}$
A particle slides from rest down a smooth slope of height $h$ and stops on a rough surface with $\mu$ : distance?: $d=\frac h\mu$ ; the mass cancels.
Does the speed of a projectile on landing depend on the angle?: No, only on the heights at start and end (ignoring air resistance).
When do you use energy instead of $F=ma$ ?: For speeds and distances; use $F=ma$ for accelerations and forces.
What is the resolved friction force on a slope?: $F=\mu R$ with $R=mg\cos\alpha$ .

Exam questions on Conservation of mechanical energy

A ball of mass $0.2$ kg is thrown from a point $1.5$ m above horizontal ground with a speed of $12$ m s $^{-1}$ . Air resistance is negligible. Take $g=9.8$ m s $^{-2}$ .
For the ball thrown vertically upwards, find its speed when it is $5$ m above the ground on the way up.2 marks
A particle of mass $2$ kg is released from rest at a point $A$ on a smooth plane inclined at $30^\circ$ to the horizontal. $A$ is $4$ m from the foot $B$ of the plane, measured along the plane. At $B$ the plane meets a rough horizontal surface with no loss of speed. The coefficient of friction between the particle and the horizontal surface is $0.4$ . Take $g=9.8$ m s $^{-2}$ .
Find the speed of the particle when it has moved $2$ m along the horizontal surface.2 marks
A child of mass $30$ kg slides from rest down a straight slide of length $6$ m, with the top of the slide $3$ m above the bottom. At the bottom the child's speed is $5$ m s $^{-1}$ . The resistance to motion is constant. Take $g=9.8$ m s $^{-2}$ .
Find the total mechanical energy lost by the child as she slides to the bottom.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).