SPH4U · Unit 2 test review

Energy and Momentum: Test Review

Everything for the Unit 2 test on one page: the key facts from all 4 lessons, the mistakes that cost the most marks, and a practice test that points you back to the lessons you need.

Key facts

Work, Energy and the Work-Energy Theorem

Work by a constant force: $W = F Δ d cos θ$ . It is a scalar in joules and is negative when the force opposes the motion.
A force perpendicular to the displacement ( $θ = 9 0^{\circ}$ ) does zero work.
Work-energy theorem: $W_{n e t} = Δ E_{k} = \frac{1}{2} m v_{2}^{2} - \frac{1}{2} m v_{1}^{2} .$
With friction: $E_{k 1} + E_{g 1} = E_{k 2} + E_{g 2} + E_{t h}$ , where $E_{t h} = F_{f} Δ d$ and $F_{f} = μ_{k} F_{N}$ .
Power: $P = W /Δ t$ , and for a constant force along the motion $P = F v$ . $1 W = 1 J/s$ .

Springs and Simple Harmonic Motion

Hooke's law: the spring force is $F_{x} = - k x$ , a restoring force proportional to the displacement from equilibrium.
Elastic potential energy: $E_{e} = \frac{1}{2} k x^{2}$ , the area under the force-extension graph.
In a frictionless mass-spring system, $\frac{1}{2} k A^{2} = \frac{1}{2} m v^{2} + \frac{1}{2} k x^{2}$ and $v_{ma x} = A k / m$ .
Period of a mass on a spring: $T = 2 π m / k$ , independent of amplitude and of $g$ .
SHM is the projection of uniform circular motion onto a straight line.
Damping converts the oscillation's mechanical energy into thermal energy, so the amplitude decreases.

Momentum and Impulse

Momentum: $p = m v$ , a vector in $kg \cdot m/s$ that points along the velocity.
Impulse-momentum theorem: $F_{n e t} Δ t = Δ p = m v_{2} - m v_{1}$ .
Impulse is the area under a force-time graph; $1 N \cdot s = 1 kg \cdot m/s$ .
Average force: $F_{a vg} = Δ p /Δ t$ . For the same $Δ p$ , a longer time means a smaller force.
A rebound reverses the velocity, so its change in momentum is larger than for stopping alone.

Collisions in One and Two Dimensions

Conservation of momentum for an isolated system: $m_{1} v_{1} + m_{2} v_{2} = m_{1} v_{1}^{'} + m_{2} v_{2}^{'} .$
Momentum is conserved in every collision and explosion; kinetic energy is conserved only in elastic collisions.
Perfectly inelastic (objects stick): $m_{1} v_{1} + m_{2} v_{2} = (m_{1} + m_{2}) v^{'} .$
Elastic, head-on, target at rest: $v_{1}^{'} = \frac{m _{1} - m _{2}}{m _{1} + m _{2}} v_{1}$ and $v_{2}^{'} = \frac{2 m _{1}}{m _{1} + m _{2}} v_{1}$ .
In two dimensions, conserve the $x$ and $y$ components of momentum separately.

Where marks get lost

Using the ramp length instead of the vertical height in $E_{g} = m g h$ . Gravitational energy depends only on height; the ramp length belongs in the friction term. (Work, Energy and the Work-Energy Theorem)
Taking the normal force as $m g$ when a rope pulls at an angle or the surface is tilted. Resolve forces first, then find $F_{f} = μ_{k} F_{N}$ . (Work, Energy and the Work-Energy Theorem)
Forgetting that work can be negative. Friction and any force opposing the motion do negative work and reduce kinetic energy. (Work, Energy and the Work-Energy Theorem)
Using the applied force alone in the work-energy theorem. It is the net work, from all the forces together, that equals the change in kinetic energy. (Work, Energy and the Work-Energy Theorem)
Counting gravity twice, once as work done by gravity and again as a change in $E_{g}$ . Choose one way to account for it. (Work, Energy and the Work-Energy Theorem)
Leaving the stretch in centimetres. Hooke's law and $\frac{1}{2} k x^{2}$ need $x$ in metres; $5 cm$ entered as $5$ makes the energy $10000$ times too big. (Springs and Simple Harmonic Motion)
Writing the elastic energy as $\frac{1}{2} k x$ or $k x^{2}$ . The force grows from zero, so the energy is the triangle area $\frac{1}{2} k x^{2}$ . (Springs and Simple Harmonic Motion)
Thinking a larger amplitude gives a longer period. For SHM the period depends only on $m$ and $k$ ; a larger amplitude just means a higher maximum speed. (Springs and Simple Harmonic Motion)
Putting $g$ into the period of a mass on a spring. $T = 2 π m / k$ holds for horizontal and vertical springs alike. (Springs and Simple Harmonic Motion)
Saying the mass is fastest at the ends because the force is largest there. The force and acceleration are largest at the ends, but the speed is zero there and greatest at equilibrium. (Springs and Simple Harmonic Motion)
Ignoring direction when an object bounces. A ball hitting a wall at $10 m/s$ and rebounding at $8 m/s$ has $Δ v = 18 m/s$ , not $2 m/s$ . (Momentum and Impulse)
Saying an airbag reduces the change in momentum. The momentum change is the same; the airbag increases the time and so reduces the force. (Momentum and Impulse)
Using the peak force from a force-time graph as if it acted the whole time. Impulse is the area under the curve, which for a triangle is half of peak times duration. (Momentum and Impulse)
Leaving the contact time in milliseconds. $1.5 ms$ is $1.5 \times 1 0^{- 3} s$ ; using $1.5$ gives a force a thousand times too small. (Momentum and Impulse)
Treating momentum as a scalar like kinetic energy. Momentum has direction, and in one dimension that means a sign. (Momentum and Impulse)
Dropping the signs of velocities in opposite directions. Two carts moving toward each other must have momenta of opposite sign. (Collisions in One and Two Dimensions)
Assuming kinetic energy is conserved in every collision. Only momentum is always conserved; check kinetic energy before calling a collision elastic. (Collisions in One and Two Dimensions)
Using one mass instead of the combined mass for objects that stick together after a perfectly inelastic collision. (Collisions in One and Two Dimensions)
Adding the speeds of two-dimensional momenta directly. Resolve into components, conserve $x$ and $y$ separately, then recombine. (Collisions in One and Two Dimensions)
Thinking an explosion from rest has no momentum afterward because the pieces are moving. Each piece has momentum; the vector total is zero. (Collisions in One and Two Dimensions)

Practice test

24 questions from across the unit. Each one shows the lesson it comes from.

Work, Energy and the Work-Energy Theorem
1
A child pulls a sled $8.0 m$ along level snow with a force of $60 N$ directed $4 0^{\circ}$ above the horizontal. How much work does the child do, in joules? Give three significant figures.
J
Work, Energy and the Work-Energy Theorem
2
A $1200 kg$ car speeds up from $15 m/s$ to $25 m/s$ . How much net work was done on it, in kilojoules?
kJ
Work, Energy and the Work-Energy Theorem
3
A frictionless roller coaster car starts from rest at the top of a hill and drops a vertical distance of $20 m$ . What is its speed at the bottom, in metres per second?
m/s
Work, Energy and the Work-Energy Theorem
4
A $70 kg$ student runs up a flight of stairs with a vertical rise of $4.5 m$ in $6.0 s$ . What average power do they develop against gravity, in watts?
W
Work, Energy and the Work-Energy Theorem
5
A $10 N$ force acts on an object that moves $2 m$ . Order these angles between force and displacement from the most positive work to the most negative work.
1. 1 $18 0^{\circ}$
2. 2 $12 0^{\circ}$
3. 3 $9 0^{\circ}$
4. 4 $0^{\circ}$
5. 5 $6 0^{\circ}$
Work, Energy and the Work-Energy Theorem
6
Two identical balls leave the edge of the same cliff at the same speed. One is thrown horizontally and one is thrown straight up. Ignoring air resistance, which is moving faster when it reaches the ground?
Springs and Simple Harmonic Motion
7
A force of $12 N$ stretches a spring by $3.0 cm$ . What is the spring constant, in N/m?
N/m
Springs and Simple Harmonic Motion
8
A mass on a spring oscillates with an amplitude of $4 cm$ . It is restarted with an amplitude of $8 cm$ . What happens to the period?
Springs and Simple Harmonic Motion
9
Where is a mass oscillating horizontally on a frictionless spring moving fastest?
Springs and Simple Harmonic Motion
10
A $0.50 kg$ mass on a spring with $k = 200 N/m$ oscillates on a frictionless surface with an amplitude of $10 cm$ . What is its maximum speed, in m/s?
m/s
Springs and Simple Harmonic Motion
11
What is the one-word name for the gradual loss of amplitude in an oscillation caused by friction or air resistance?
Springs and Simple Harmonic Motion
12
Match each point in the motion of a mass on a spring to what is true there.
At either end
At equilibrium
Halfway to an end
Every point in the cycle
Momentum and Impulse
13
What is the magnitude of the momentum of a $0.060 kg$ tennis ball moving at $50 m/s$ , in $kg \cdot m/s$ ?
kg·m/s
Momentum and Impulse
14
A constant force of $250 N$ acts on a hockey puck for $0.020 s$ . What impulse does it deliver, in $N \cdot s$ ?
N·s
Momentum and Impulse
15
A $0.20 kg$ ball hits a wall at $10 m/s$ and rebounds straight back at $8.0 m/s$ . What is the magnitude of its change in momentum, in $kg \cdot m/s$ ?
kg·m/s
Momentum and Impulse
16
Two identical eggs fall from the same height, one onto a pillow and one onto a tile floor. Which quantity is the same for both eggs as they are stopped?
Momentum and Impulse
17
Momentum is measured in $kg \cdot m/s$ . What equivalent unit, built from newtons, is usually used for impulse?
Momentum and Impulse
18
Match each safety feature to how it reduces the force on a person.
Crumple zone
Helmet foam liner
Seat belt
Bending your knees on landing
Collisions in One and Two Dimensions
19
A $3.0 kg$ cart moving at $4.0 m/s$ hits a $1.0 kg$ cart at rest, and they stick together. What is their speed afterward, in m/s?
m/s
Collisions in One and Two Dimensions
20
Which quantity is conserved in every collision between two objects with no net external force?
Collisions in One and Two Dimensions
21
A $4.0 kg$ rifle at rest fires a $10 g$ bullet at $400 m/s$ . What is the rifle's recoil speed, in m/s?
m/s
Collisions in One and Two Dimensions
22
A $2.0 kg$ cart moving at $6.0 m/s$ collides with a $1.0 kg$ cart at rest, and they stick. How much kinetic energy is lost, in joules?
J
Collisions in One and Two Dimensions
23
Which particle was proposed in 1930 to explain the energy and momentum that seemed to go missing in beta decay?
Collisions in One and Two Dimensions
24
Match each event to the type of collision or interaction.
Two air-track gliders bounce off spring bumpers with no kinetic energy lost
A dart sticks in a hanging wooden block
A tennis ball bounces off a racket, losing some kinetic energy
Two skaters at rest push apart

Endless drill

Fresh numbers every time, for the calculation questions in this unit.

Endless drillnew numbers every time

Generating a problem…