**Question**

The total mass of a cyclist and her bicycle is 75 kg. The cyclist ascends a straight hill of length 0.7 km inclined at 1.5Å to the horizontal. Her speed at the bottom of the hill is 10 \(ms^{-1}\) and at the top it is 5\( ms^{-1}\)

. There is a resistance to motion, and the work done against this resistance as the cyclistascends the hill is 2000 J. The cyclist exerts a constant force of magnitude F N in the direction of motion. Find F.

**Answer/Explanation**

Initial KE=\(\frac{1}{2}\times 75\times 10^{2}\)

final KE=\(\frac{1}{2}\times 75\times 5^{2}\)

PE gained = × 75 700 sin1.5 g [=13 743] WD by

F= F × 700

WD by F + Initial KE = Final KE + PE gain + 2000 F

= 18.5

### Question

A tractor of mass 3700 kg is travelling along a straight horizontal road at a constant speed of 12 \(m s^{−1}\). The total resistance to motion is 1150 N.

(i) Find the power output of the tractor’s engine. The tractor comes to a hill inclined at 4Å above the horizontal. The power output is increased to 25 kW and the resistance to motion is unchanged.

(ii) Find the deceleration of the tractor at the instant it begins to climb the hill.

(iii) Find the constant speed that the tractor could maintain on the hill when working at this power.

**Answer/Explanation**

(i) Power = 1150 × 12 = 13 800W

(ii) Driving force = \(\frac{25000}{12}\)

\(\frac{25000}{12}\)– 1150 – 3700g sin 4 = 3700a

a = –0.445 m \(s^{–2}\)

2(iii)\( \frac{25000}{v}\)– 1150 – 3700gsin 4 = 0

v = 6.70 m \(s^{-1}\)

**Question**

A car of mass 800 kg is moving up a hill inclined at 1Å to the horizontal, where sin 1 = 0.15. The initial speed of the car is 8\( m s^{−1}\). Twelve seconds later the car has travelled 120 m up the hill and hasspeed 14\( m s^{−1}\).

(i) Find the change in the kinetic energy and the change in gravitational potential energy of the car.

(ii) The engine of the car is working at a constant rate of 32 kW. Find the total work done against the resistive forces during the twelve seconds.5(ii)

**Answer/Explanation**

(i) KE gain = 1⁄2 × 800 ×\( (14^{2} – 8^{2}) \)= 52800 J

PE gain = 800 × 10 × 120 × 0.15 = 144000 J

(ii) WD by engine = 32000 × 12

32000 × 12 = 144000 + 52800 + WD against F

WD against F = 187200 J

The total mass of a cyclist and her bicycle is 70 kg. The cyclist is riding with constant power of 180W up a straight hill inclined at an angle α to the horizontal, where sin α = 0.05. At an instant when the cyclist’s speed is 6 m s^{−1} , her acceleration is −0.2m s^{−2}. There is a constant resistance to motion of magnitude F N.

### (a) Question

Find the value of F.

**Answer/Explanation**

Ans:

Forward force exerted by cyclist driving force = \(\frac{180}{6}\left [ =30N \right ]\)

DF – F – 70g sin α = 70 × ‒ 0.2

30 – F – 70g × 0.05 = 70 × ‒ 0.2

F = 9

### (b) Question

Find the steady speed that the cyclist could maintain up the hill when working at this power.

**Answer/Explanation**

Ans:

\(\frac{180}{v}\) – F – 70g × sin α = 0

v = 4.09 m s^{−1}

A car of mass mkg is towing a trailer of mass 300 kg down a straight hill inclined at 3^{0} to the horizontal at a constant speed. There are resistance forces on the car and on the trailer, and the total work done against the resistance forces in a distance of 50 m is 40 000 J. The engine of the car is doing no work and the tow-bar is light and rigid.

### (a) Question

Find the value of m.

The resistance force on the trailer is 200 N.

**Answer/Explanation**

Ans:

PE lost in 50 m = (m + 300) g × 50 sin 3

(m + 300) g × 50 sin 3 – 40 000 = 0

m = 1230 to 3 sf

Alternative method for question 3(a)

Resistance force R = \(\frac{40000}{50}\left [ =800N \right ]\)

(m + 300)g sin 3 – R = 0

m = 1230 to 3 sf

### (b) Question

Find the tension in the tow-bar between the car and the trailer.

**Answer/Explanation**

Ans:

T + 300 g sin 3 – 200 = 0 (Trailer)

or

mg sin 3 = T + 600 (Car)

T = 43[.0] N to 3 sf

### Question:

A crane is used to raise a block of mass 600 kg vertically upwards at a constant speed through a height of 15 m. There is a resistance to the motion of the block, which the crane does 10 000 J of work to overcome.

### (a) Question

Find the total work done by the crane.

**Answer/Explanation**

Ans:

600g × 15 [= 90 000]

Total work done by crane = [90 000 + 10 000 =] 100 000 J

### (b) Question

Given that the average power exerted by the crane is 12.5 kW, find the total time for which the block is in motion.

**Answer/Explanation**

Ans:

100 000 = 12 500 × t

Time = 8 s

Alternative scheme for question 1(b)

Average force F = \(\frac{Total WD}{15}\)

Average velocity v = \(\frac{s}{t} = \frac{15}{t}\)

\(P = Fv \rightarrow 12500 = \frac{Total WD}{15}\times \frac{15}{t}\)

Time = 8 s

### Question:

A car of mass 1600 kg travels at constant speed 20 m s^{−1} up a straight road inclined at an angle of sin^{−1 }0.12 to the horizontal.

### (a) Question

Find the change in potential energy of the car in 30 s.

**Answer/Explanation**

Ans:

s = × 30 20

PE change = 1600 × g × s × 0.12

[PE change =1600 × g × 20 × 30 × 0.12 g ]

Change in PE = 1152000 J

### (b) Question

Given that the total work done by the engine of the car in this time is 1960 kJ, find the constant force resisting the motion.

**Answer/Explanation**

Ans:

1960 000 = WD_{res} + their PE

[1960 000 = WD_{res} + 1152 000]

[WD_{res} =80800 J]

R = WD_{res }÷ 600

Force resisting motion R = 1350 N to 3sf

### (c) Question

Calculate, in kW, the power developed by the engine of the car.

**Answer/Explanation**

Ans:

\(P = \left ( \frac{4040}{3} + 1600 \times g \times 0.12 \right )\times 20\)

\(\left [ = \frac{196000}{3} \right ]\)

P= 65.3 kW

### (d) Question

Given that this power is suddenly decreased by 15%, find the instantaneous deceleration of the car.

**Answer/Explanation**

Ans:

\(0.85 \times \frac{196000}{3}= DF\times 20\)

\(DF – R – 1600g \times 0.12 = 1600a\)

\(\left [ \frac{8330}{3} – \frac{4040}{3} – 1920 = 1600a \right ]\)

\(a = \left [ – \right ] 0.306ms^{-2}\)

### Question

A slide in a playground descends at a constant angle of 30^{o} for 2.5 m. It then has a horizontal section in the same vertical plane as the sloping section. A child of mass 35 kg, modelled as a particle P, starts from rest at the top of the slide and slides straight down the sloping section. She then continues along the horizontal section until she comes to rest (see diagram). There is no instantaneous change in speed when the child goes from the sloping section to the horizontal section.

The child experiences a resistance force on the horizontal section of the slide, and the work done against the resistance force on the horizontal section of the slide is 250 J per metre.

(a) It is given that the sloping section of the slide is smooth.

(i) Find the speed of the child when she reaches the bottom of the sloping section. [3]

(ii) Find the distance that the child travels along the horizontal section of the slide before she

comes to rest. [2]

(b) It is given instead that the sloping section of the slide is rough and that the child comes to rest on

the slide 1.05 m after she reaches the horizontal section.

Find the coefficient of friction between the child and the sloping section of the slide. [6]

**Answer/Explanation**

(a) (i) PE = 35 × 2.5sin30

\(\frac{1}{2}\times 35v^{2}=35g\times 2.5\sin 30\)

v= 5 m s^{–1}

**Alternative method for Question (a)(i)**

mg sin30= ma leading to a= 5

v = 0 + 2 × 5 × 2.5

v= 5 m s^{–1}

7 (a) (ii) \(\frac{1}{2}\times 35\times 5^{2}=250d\)

d =1.75 m

** Alternative method for Question (a)(ii)**

\(-250=35a\ leading \ to \ a=-\frac{50}{7}=-7.14\)

0 = 5^{2} + 2 (a) (d)

d =1.75 m

(b) \(\frac{1}{2}\times 35v^{2}=250\times 1.05[v^{2}=15]\)

or

\(-250=35a\ leading\ to\ a=-\frac{50}{7}\)

\(O=v^{2}+2\times -\frac{50}{7}\times 1.05 \ [v^{2}=15]\)

R = 35g cos30 [303.11]

v^{2} = 0 + 2 × a × 2.5 = 15 leading to a = 3

or

PE change = = 35g × 2.5sin 30 [437.5]

35 sin 30 − F = 35a = or [175 − F = 35a]

or

35 × 2.5sin 30 F × 2.5 + \(\frac{1}{2}\times 35\times 15 [437.5=F\times 2.5+262.5]\)

F = μ × R

μ = 0.231

** (b) Alternative method for Question (b)**

R = 35g cos 30

PE change = = 35g × 2.5sin 30 [437.5 ]

WD against friction on the flat = 250 × 1.5

35g × 2.5sin 30 F × 2.5 + 250 × 1.05 = [437.5 × F 2.5 + 262.5 ]

F = μ × R

μ = 0.231

**Question**

A car of mass 1400 kg is travelling at constant speed up a straight hill inclined at \(\alpha\) to the horizontal, where \(sin\alpha=0.1\). There is a constant resistance force of magnitude 600 N. The power of the car’s engine is 22500 W.

(a) Show that the speed of the car is \(11.25 ms^{-1}\).

The car, moving with speed \(11.25ms^{-1}\), comes to a section of the hill which is inclined at \(2^o\) to the horizontal.

(b) Given that the power and resistance force do not change, find the initial acceleration of the car up this section of the hill.

**Answer/Explanation**

**Ans:**

- Driving force = DF = \(\frac{22500}{v}\)

DF – 1400g \(\times\) 0.1 – 600 = 0\)

\(v=11.25ms^{-1}\) - DF-1400g sin2 – 600 = 1400a

\(\frac{22500}{11.25}-1400g sin 2 -600=1400a\)

\(a=0.651ms^{-2}\) (3sf)

### Question

A car of mass 1500 kg is pulling a trailer of mass 750 kg up a straight hill of length 800 m inclined at

an angle of sin^{−1} 0.08 to the horizontal. The resistances to the motion of the car and trailer are 400 N

and 200 N respectively. The car and trailer are connected by a light rigid tow-bar. The car and trailer

have speed 30 m s^{−1} at the bottom of the hill and 20 m s^{−1 }at the top of the hill.

(a) Use an energy method to find the constant driving force as the car and trailer travel up the hill. [5]

After reaching the top of the hill the system consisting of the car and trailer travels along a straight

level road. The driving force of the car’s engine is 2400 N and the resistances to motion are unchanged.

(b) Find the acceleration of the system and the tension in the tow-bar. [4]

**Answer/Explanation**

Ans

(a) KE (final) = ½ × 1500 × 20^{2} + ½ × 750 × 20^{2}

KE (initial) = ½ × 1500 × 30^{2} + ½ × 750 × 30^{2}

PE gain = 2250 × 10 × 800 × 0.08

WD against friction = 600 × 800

½ × 2250 × 30^{2} + DF × 800 = 600 × 800

+ ½ × 2250 × 20^{2} + 2250 × 10 × 800 × 0.08

DF = 1700 N

(b) 2400 – 600 = 2250*a*

or

T – 200 = 750a **and** 2400 – 400 – T = 1500*a*

Attempting to solve for *a* or for T

T = 800 N and a = 0.8 ms^{–2}

### Question

A car of mass 1400 kg is moving along a straight horizontal road against a resistance of magnitude

350 N.

(a) Find, in kW, the rate at which the engine of the car is working when it is travelling at a constant

speed of 20 m s^{−}1. [2]

(b) Find the acceleration of the car when its speed is 20 m s^{−1 }and the engine is working at 15 kW. [3]

**Answer/Explanation**

(a) P = 350 × 20

P = 7 kW

2 (b) 15 000 = DF × 20 [DF = 750]

DF – 350 = 1400*a*

\(a=\frac{2}{7}ms^{-2}\)

### Question

A child of mass 35 kg is swinging on a rope. The child is modelled as a particle *P* and the rope

is modelled as a light inextensible string of length 4 m. Initially *P* is held at an angle of 45^{o} to the

vertical (see diagram).

(a) Given that there is no resistance force, find the speed of *P* when it has travelled half way along

the circular arc from its initial position to its lowest point. [4]

(b) It is given instead that there is a resistance force. The work done against the resistance force

as P travels from its initial position to its lowest point is *X J*. The speed of P at its lowest point

is 4 m s^{−1}.

Find *X*. [3]

**Answer/Explanation**

Ans

(a) Attempt at finding PE lost

PE lost = 35 g (4cos22.5 – 4cos45)

\(\frac{1}{2}\times 35v^{2}=35g(4\cos 22.5-4\cos 45)\)

Speed = 4.16 ms^{–1} (4.1643…)

(b) Use of the work-energy equation in the form: PE lost = KE gain + WD against resistance

\(\frac{1}{2}\times 35\times 4^{2}=35g(4-4\cos 45)=X\)

X = 130 (130.05…)

### Question

A car of mass 1800 kg is towing a trailer of mass 400 kg along a straight horizontal road. The car and

trailer are connected by a light rigid tow-bar. The car is accelerating at 1.5 m s^{−2}. There are constant

resistance forces of 250 N on the car and 100 N on the trailer.

(a) Find the tension in the tow-bar.

**(b) Find the power of the engine of the car at the instant when the speed is 20 m s ^{−1}. **

**Answer/Explanation**

(a) [T – 100 = 400 × 1.5]

T = 700 N

(b)F – 250 – 100 = 2200 × 1.5 (F = 3650 N)

(**M1** for using Newton’s second law for the system or for the car using the result from **2(a))**

For use of power =*Fv*

73 000 W or 73 kW

**Question**

The diagram shows the vertical cross-section of a surface. A, B and C are three points on the cross-section. The level of B is h m above the level of A. The level of C is 0.5m below the level of A. A particle of mass 0.2 kg is projected up the slope from A with initial speed \(5ms^{-1}\). The particle remains in contact with the surface as it travels form A to C.

(a) Given that the particle reaches B with a speed of \(3ms^{-1}\) and that there is no resistance force.

find h.

(b) It is given instead that there is a resistance force and that the particle does 3.1J of work against the resistance force as it travels from A to C.

**Answer/Explanation**

**Ans:**

(a) Initial KE \(=1/2 \times 0.2 \times 5^2\)

or Final KE \(=1/2 \times 0.2 \times 3^2\)

\(1/2 \times 0.2 \times 5^2 = 0.2gh + 1/2 \times 0.2 \times 3^2\)

h = 0.8

(b) Apply work-energy equation form A to C

\(1/2 \times 0.2 \times 5^2 – 3.1 + 0.2g \times 0.5 = 1/2 \times 0.2v^2\)

Speed = \(2ms^{-1}\)

**Question**

A lorry mass 16 000 kg is travelling along a straight horizontal road. The engine of the lorry is working at constant power. The work done by the driving force in 10s is 750 000 J.

(a) Find the power of the lorry’s engine.

(b) There is a constant resistance force acting on the lorry of magnitude 2400 N.

Find the acceleration of the lorry at an instant when its speed is 25ms^{-1}.

**Answer/Explanation**

**Ans:**

(a) Power = 750000/10 = 75000 W or 75 kW

(b) Driving force DF = 75000/25

[DF-2400=16000a]

\(a=0.0375ms^{-2}\)

**Question**

A lorry has mass 12 000 kg.**(i)** The lorry moves at a constant speed of \(5m^{-1}\) up a hill inclined at an angle of 1 to the horizontal,

where sin 1 = 0.08. At this speed, the magnitude of the resistance to motion on the lorry is 1500 N. Show that the power of the lorry’s engine is 55.5 kW. When the speed of the lorry is \(ms^{-1}\) the magnitude of the resistance to motion is \(kv^{2}\) where k is aconstant.**(ii)** Show that k = 60.

**(iii)** The lorry now moves at a constant speed on a straight level road. Given that its engine is still

working at 55.5 kW, find the lorry’s speed.

**Answer/Explanation**

**3(i)** DF = 1500+ 12 000×g× 0.08 [DF = 11100]

Power = DF× 5 Power = 11 100× 5 = 55.5 kW

**3(ii)** \(k×5^{2}\)= 1500,k = 60

**3(iii)** DF = \(60\nu ^{2}\)

55500 = DF×v = \(60\nu ^{2}\)×v =\( 60\nu ^{3}\)

v = 9.74 \(ms^{-1}\)

**Question**

The diagram shows the vertical cross-section PQR of a slide. The part PQ is a straight line of length 8 m inclined at angle ! to the horizontal, where sin ! = 0.8. The straight part PQ is tangential to the curved part QR, and R is h m above the level of P. The straight part PQ of the slide is rough and the curved part QR is smooth. A particle of mass 0.25 kg is projected with speed 15 m \(s^{-1}\) from P towards Q and comes to rest at R. The coefficient of friction between the particle and PQ is 0.5.**(i)** Find the work done by the friction force during the motion of the particle from P to Q.

**(ii)** Hence find the speed of the particle at Q.

**(iii ****)**Find the value of h.

**Answer/Explanation**

**7(i) **R = 0.25g × 0.6 [= 1.5]

[F = 0.5 × 0.25g × 0.6] [F = 0.75] [WD against friction =F × 8]WD = 6 J

**7(ii)** [1⁄2 × 0.25 ×\(15v^{-2}\) =1⁄2 × 0.25 ×\( v^{-2} \)+ 6 + 0.25g × 8 × 0.8] v = 7m \(s^{-1}\) [–F – 0.25g sinα = 0.25a] a = –11

\(ms^{-1}\) v = 7 \(ms^{-1}\)

**7(iii)** [1⁄2 × 0.25 ×\( 7^{2}\) = 0.25 ×g ×H] Or

[1⁄2 ×m × \(7^{2}\) =m ×g ×H] H =\( 7^{2}/2g\) = 2.45 m Total height

h = 6.4 +H = 8.85 [1⁄2 × 0.25 ×\(15^{2}\) = 6 + 0.25g ×h] h = 8.85

**Question**

A car of mass 1500 kg is pulling a trailer of mass 300 kg along a straight horizontal road at a constant

speed of 20 m\( s^{-1}\). The system of the car and trailer is modelled as two particles, connected by a light rigid horizontal rod. The power of the car’s engine is 6000 W. There are constant resistances to motion of R N on the car and 80 N on the trailer.**(i)** Find the value of R.

The power of the car’s engine is increased to 12 500 W. The resistance forces do not change.**(ii)** Find the acceleration of the car and trailer and the tension in the rod at an instant when the speed of the car is 25 \(ms^{-1}\)

**Answer/Explanation**

**4(i)** Driving force = 6000/20 [= 300 N] R = 300 – 80 = 220

**4(ii)** [New driving force DF = 12500/25 = 500 N

Car: DF –T –R = 1500

a Trailer: T – 80 = 300a System: DF – 80 –R = 1800 a] Two correct equations a = 0.111\(ms^{-2}\) T = 113 N (= 113.3333…)

**Question**

A van of mass 2500 kg descends a hill of length 0.4 km inclined at 4Å to the horizontal. There is a

constant resistance to motion of 600 N and the speed of the van increases from \( 20ms^{-1}\)to \(30 ms^{-1}\)as it descends the hill. Find the work done by the van’s engine as it descends the hill.

**Answer/Explanation**

[KE gained \frac{1}{2}\times 2500(30^{2}-20^{2})(=625000j)

PE lost =× = 2500 400sin 4 (-697564.7j)

[WD by engine +2500g\times 400sin+\frac{1}{2}\times 2500\times 20^{2}

=600\times 400+\frac{1}{2}\times 25000\times 30^{2} Work done by engine

+ PE lost = 600× 400+ 625 000

Work done = 167 000 J (167 435.2…)

**Question**

A high-speed train of mass 490 000 kg is moving along a straight horizontal track at a constant speed of\( 85 ms^{-1}\). The engines are supplying 4080 kW of power.**(i)** Show that the resistance force is 48 000 N.

**(ii)** The train comes to a hill inclined at an angle \Theta \AA above the horizontal, where sin \Theta \AA =\(\frac{1}{200}\) . Given that the resistance force iunchanged, find the power required for the train to keep moving at the same constant speed of\( 85 ms^{-1}\).

**Answer/Explanation**

**(i)** Resistance = Driving force=\(\frac{4080000}{85}=48000N**(ii)**DF=\frac{p}{85}

DF-48000-490000g\times \frac{1}{200}=0

P=72500\times 85=6.16MW\)

### Question

A car of mass 1200 kg has a greatest possible constant speed of 60 m s^{−1} along a straight level road. When the car is travelling at a speed of v \(m s^{−1}\) there is a resistive force of magnitude 35v N.**(i)** Find the greatest possible power of the car.

**(ii)** The car travels along a straight level road. Show that, at an instant when its speed is 30\( m s^{−1}\),the greatest possible acceleration of the car is \(2.625 m s^{−2}\).

**(iii)** The car travels at a constant speed up a hill inclined at an angle of \(sin^{-1}\frac{7}{48}\) to the horizontal.

Find the greatest possible speed of the car.

**Answer/Explanation**

**(i)** Driving force = 35 × 60

Power =\( 35 × 60^{2}\) = 126000 W

**(ii)** Driving force is Dm =\(\frac{12600}{30}\)

DF -35 \times 30 = 1200 a

** (iii)** DF= \(\frac{12600}{v}\)

\(35V^{2}+1750v-126000\)=0

OR\( v^{2}+50v-3600=0

v=40 ms^{-1}\)

### Question

A girl, of mass 40 kg, slides down a slide in a water park. The girl starts at the point A and slides to the point B which is 7.2 metres vertically below the level of A, as shown in the diagram.**(i)** Given that the slide is smooth and that the girl starts from rest at A, find the speed of the girl at B.

**(ii)** It is given instead that the slide is rough. On one occasion the girl starts from rest at A and reaches B with a speed of 10 \(m s^{−1}\)

. On another occasion the girl is pushed from A with an initial

**Answer/Explanation**

speed V \(m s^{−1}\)

and reaches B with speed 11 \(m s^{−1}\)

. Given that the work done against friction is the same on both occasions, find V.

**(i) ** \(\frac{1}{2}\times 40\times v^{2}=40\times g \times 7.2

v=12ms^{1}\)

**(ii)** Work done against friction(WDF)

WDF= \(40 \times g \times 7.2-\frac{1}{2}\times 40 \times 10^{2}[=880]

\frac{1}{2} \times 40 \times v^{2}+40 \times g \times 7.2=\frac{1}{2} \times 40 \times11^{2}+880\)

**Question**

A car of mass 900 kg is moving on a straight horizontal road ABCD. There is a constant resistance of magnitude 800 N in the sections AB and BC, and a constant resistance of magnitude R N in the

section CD. The power of the car’s engine is a constant 36 kW.**(i)** The car moves from A to B at a constant speed in 120 s. Find the speed of the car and the distance AB.

The car’s engine is switched off at B.**(ii)** The distance BC is 450 m. Find the speed of the car at C.

**(iii)** The car comes to rest at D. The distance AD is 6637.5 m. Find the deceleration of the car and the value of R.

**Answer/Explanation**

**(i)** 36000 = 800v

v =\( 45ms^{-1}\)

AB = 45 × 120 = 5400m

**(ii)** −800 = 900a [a = –8/9]

\(v^{2}\)\(=45^{2}-\frac{16}\)\({9}\times 450\)

\(v=35ms^{-1}\)

**(iii)** CD = 6637.5 – 5400 – 450 = 787.5

\(0=-35^{2}-2d\times 787.5

d=7/9=0.77ms^{-2}\)

\(p=900\times (7/9)=700\)

### Question

A constant resistance of magnitude 1350 N acts on a car of mass 1200 kg.

** (i)** The car is moving along a straight level road at a constant speed of 32 m s^{−1}. Find, in kW, the rate at which the engine of the car is working. [2]

**(ii)** The car travels at a constant speed up a hill inclined at an angle of ∅ to the horizontal, where sin ∅ = 0.1, with the engine working at 76.5 kW. Find this speed. [3]

**Answer/Explanation**

Ans:

**(i)** DF = 1350

*P* = 1350 × 32 = 43.2 kW

** (ii)** DF – 1350 – 1200g × 0.1 = 0

[DF = 2550]

DF = 76500/*v*

*v* = 30ms^{–1}

### Question

A lorry of mass 24 000 kg is travelling up a hill which is inclined at 3° to the horizontal. The power developed by the lorry’s engine is constant, and there is a constant resistance to motion of 3200 N.

** (i)** When the speed of the lorry is 25 m s^{−1}, its acceleration is 0.2 m s^{−2}. Find the power developed by the lorry’s engine. [4]

** (ii)** Find the steady speed at which the lorry moves up the hill if the power is 500 kW and the resistance remains 3200 N. [2]

**Answer/Explanation**

**(i)** F – 24000g sin 3 – 3200 =

24000 × (0.2)

Power = *Fv* = 20561 × 25

Power = 514kW

**(ii)** DF = 3200 + 24000g sin 3

[=15761]

*v* = 500000 /15761 = 31.7ms^{–1}

### Question

A weightlifter performs an exercise in which he raises a mass of 200 kg from rest vertically through a distance of 0.7 m and holds it at that height.

** (i)** Find the work done by the weightlifter. [2]

** (ii)** Given that the time taken to raise the mass is 1.2 s, find the average power developed by the weightlifter. [2]

**Answer/Explanation**

** (i) **200g × 0.7

** **** **** **** **** **Work done = 1400 J

** (ii) **1400 /1.2

** **** **** **** **** **Average Power = 1170W

### Question

A cyclist and her bicycle have a total mass of 84 kg. She works at a constant rate of P W while moving on a straight road which is inclined to the horizontal at an angle θ, where sin θ = 0.1. When moving uphill, the cyclist’s acceleration is 1.25 m s^{−2 }at an instant when her speed is 3 m s^{−1}. When moving downhill, the cyclist’s acceleration is 1.25 m s^{−2 }at an instant when her speed is 10 m s^{−1}. The resistance to the cyclist’s motion, whether the cyclist is moving uphill or downhill, is R N. Find the values of P and R. [8]

**Answer/Explanation**

**Question**

A block of mass 60 kg is pulled up a hill in the line of greatest slope by a force of magnitude 50 N acting at an angle !Å above the hill. The block passes through points A and B with speeds 8.5\( ms^{-1}\) and \(3.5 ms^{-2}\) respectively (see diagram). The distance AB is 250 m and B is 17.5 m above the level of A. The resistance to motion of the block is 6 N. Find the value of \(\alpha \)

**Answer/Explanation**

PE change = 60 g × 17.5 or

KE change = 1⁄2 60(8.52– 3.52)

[PE = 10500]

KE change = 1⁄2 60(8.52– 3.52) or

PE change = 60 g × 17.5

[KE = 1800]

WD against resistance = 6 × 250 [= 1500]

WD by pulling force =50 cosα × 250

WD = 10500 – 1800 + 1500

10200 J or 10.2 kJ

For using WD = Fdcosα

10200 = 50 × 250 cosα

α = 35.3

**Question**

A car of mass 800 kg is moving on a straight horizontal road with its engine working at a rate of

22.5 kW. Find the resistance to the car’s motion at an instant when the car’s speed is \(18 ms^{-1}\) and its acceleration is\( 1.2 ms^{-2}\)

**Answer/Explanation**

DF – R = 800 × 1.2

DF = 22500/18 [ = 1250]

Resistance is 290 N

.

**Question**

A car of mass 1100 kg starts from rest at O and travels along a road OAB. The section OA is straight, of length 1760 m, and inclined to the horizontal with A at a height of 160 m above the level of O. The section AB is straight and horizontal (see diagram). While the car is moving the driving force of the car is 1800 N and the resistance to the car’s motion is 700 N. The speed of the car is \(v m s^{-1}\)when the car has travelled a distance of x m from O.

**(i)** For the car’s motion from O to A, write down its increase in kinetic energy in terms of v and its increase in potential energy in terms of x. Hence find the value of k for which \(kv^{2}\) = x for 0 ≤ x ≤ 1760.

**Answer/Explanation**

**(ii)** Show that\( v^{2}\)= 2x − 3200 for x ≥ 1760.

**(i)** KE gain = \(550v^{2}\)

PE gain = 1000x

\([1800x=550v^{2}+1000x+700x]\)

**(ii) **At A \(5.5v^{2}=1760\rightarrow v^{2}=320\)

\(550(v^{2}-320)\)=1800(x-1760)-700(x-1760)

\(v^{2}=2x-3200 \)(cw0)

**Alternative for part (ii)**

[1800 – 700 = 1100a and \(5.5v^{2}=1760]\)

a = 1 and\( v^{2}=320

[v^{2}=320+2\times 1\times (x-1760)]

v^{2}=2x-3200\)

**Question**

A train is moving at constant speed\( V ms^{-1}\) along a horizontal straight track. Given that the power of the train’s engine is 1330 kW and the total resistance to the train’s motion is 28 kN, find the value of V.

**Answer/Explanation**

DF = 28000 B1

[1330 000 = 28000V] = V = 47.5

### Question

A lorry of mass 15 000 kg climbs from the bottom to the top of a straight hill, of length 1440 m, at a constant speed of 15 m s^{−1}. The top of the hill is 16 m above the level of the bottom of the hill. The resistance to motion is constant and equal to 1800 N.

** (i)** Find the work done by the driving force. [4]

** **On reaching the top of the hill the lorry continues on a straight horizontal road and passes through a point P with speed 24 m s^{−1}. The resistance to motion is constant and is now equal to 1600 N. The work done by the lorry’s engine from the top of the hill to the point P is 5030 kJ.

** (ii)** Find the distance from the top of the hill to the point P. [3]

**Answer/Explanation**

Ans:

**(i)** Gain in PE =15000g × 16

WD against resistance = 1800 × 1440

Work done is 4.99×10^{6} J

** (ii) **5030 000 = ½ 15 000(24^{2} – 15^{2} ) + 1600d

Distance is 1500 m

### Question

A car of mass 1250 kg travels from the bottom to the top of a straight hill of length 600 m, which is inclined at an angle of 2.5° to the horizontal. The resistance to motion of the car is constant and equal to 400 N. The work done by the driving force is 450 kJ. The speed of the car at the bottom of the hill is 30 m s^{−1}. Find the speed of the car at the top of the hill. [5]

**Answer/Explanation**

Ans:

**Special Ruling** for candidates who assume, without justification, that the driving force (DF) is constant (maximum mark 4).

**Question**

A car of mass 1200 kg moves in a straight line along horizontal ground. The resistance to motion of the car is constant and has magnitude 960 N. The car’s engine works at a rate of 17 280 W.**(i)** Calculate the acceleration of the car at an instant when its speed is 12 m\( s^{−1}\) The car passes through the points A and B. While the car is moving between A and B it has constant

speed V m\( s^{−1}\)**(ii)** Show that V = 18.

At the instant that the car reaches B the engine is switched off and subsequently provides no energy. The car continues along the straight line until it comes to rest at the point C. The time taken for the

car to travel from A to C is 52.5 s.**(iii)** Find the distance AC.

**Answer/Explanation**

**(i)** DF = 17280/12 (= 1440 N)

[DF – R = ma 1440 – 960 = 1200a]

Acceleration is 0.4 m\(s^{–2}\)

**(ii)** [17280/V – 960 = 0]

V = 18 m/s

**Question**

A car of mass 880 kg travels along a straight horizontal road with its engine working at a constant rate of P W. The resistance to motion is 700 N. At an instant when the car’s speed is 16 m \(s^{−1} \)its acceleration is 0.625 m\( s^{−2}\). Find the value of P.

**Answer/Explanation**

DF – 700 = 880 × 0.625

[P = 1250 × 16] P = (DF)v

P = 20 000

**Question**

A load of mass 160 kg is pulled vertically upwards, from rest at a fixed point O on the ground, using a winding drum. The load passes through a point A, 20 m above O, with a speed of \(1.25 m s^{−1}\)(see diagram). Find, for the motion from O to A,**(i)** the gain in the potential energy of the load, **(ii)** the gain in the kinetic energy of the load. The power output of the winding drum is constant while the load is in motion.**(iii) Given that the work done against the resistance to motion from O to A is 20 kJ and that the time taken for the load to travel from O to A is 41.7 s, find the power output of the winding drum.**

**Answer/Explanation**

**(i)** PE gain is 32 000 J **(ii)** [KE gain = 1⁄2 160 × \(1.25^{2}\)]

KE gain is 125 J

**(iii)** WD by drum = 32 000 + 125 + 20 000

[P = 52 125 ÷ 41.7]

Power is 1250 W