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Lesson Worksheet: Linear Motion with Derivatives Mathematics

In this worksheet, we will practice using differentiation to find the instantaneous velocity, speed, and acceleration of a particle.

Q1:

A particle starts from a fixed point 𝑂 and moves in a straight line. The distance traveled at time 𝑑 seconds is given by 𝑠=ο€Όπ‘‘βˆ’14πœ‹(2πœ‹π‘‘)+14πœ‹οˆcosm, where 𝑑β‰₯0.

Find an expression for the speed of the particle at time 𝑑 seconds.

  • Aο€Ό1+12(2πœ‹π‘‘)+14πœ‹οˆsin m/s
  • Bο€Ό1+12(2πœ‹π‘‘)cos m/s
  • Cο€Ό1βˆ’12(2πœ‹π‘‘)sin m/s
  • Dο€Ό1+12(2πœ‹π‘‘)sin m/s
  • Eο€Ό1βˆ’12(2πœ‹π‘‘)cos m/s

Find the maximum speed of the particle.

  • Aο€Ό32+14πœ‹οˆ m/s
  • Bο€Ό12+14πœ‹οˆ m/s
  • C12 m/s
  • D1 m/s
  • E32 m/s

Q2:

A particle of mass 2 kg moves on the positive π‘₯-axis. At time 𝑑 seconds, the particle’s displacement, 𝑠, from the origin is given by 𝑠=ο€½4𝑑+14π‘’ο‰οŽ’οŽ‘οŠ±οŠ©οm, where 𝑑β‰₯0.

Find the velocity of the particle when 𝑑=1, giving your answer to 3 decimal places.

The particle is acted upon by a force of variable magnitude, 𝐹 N, which acts in the direction of the positive π‘₯-axis.

Find the value of 𝐹 when 𝑑=2, giving your answer to 3 decimal places.

Q3:

A particle P moves in a straight line. At time 𝑑 seconds, where 𝑑β‰₯0, the acceleration of P is π‘Ž m/s2 and the velocity 𝑣 m/s is given by 𝑣=4+2(π‘˜π‘‘)sin, where π‘˜ is a constant. The initial acceleration of P is 10 m/s2.

Find the value of π‘˜.

  • A3
  • B5
  • C4
  • D15
  • E13

Using the value of π‘˜ found in part 1, work out, in terms of πœ‹, the values of 𝑑 in the interval 0β‰€π‘‘β‰€πœ‹3 for which π‘Ž=0.

  • A𝑑=πœ‹10s and 𝑑=πœ‹2s
  • B𝑑=πœ‹10s and 𝑑=πœ‹3s
  • C𝑑=3πœ‹10s and 𝑑=πœ‹2s
  • D𝑑=πœ‹3s and 𝑑=πœ‹2s
  • E𝑑=πœ‹10s and 𝑑=3πœ‹10s

Find the maximum velocity and maximum acceleration.

  • AMaximum velocity = 6 m/s, maximum acceleration = 10 m/s2
  • BMaximum velocity = 2 m/s, maximum acceleration = 10 m/s2
  • CMaximum velocity = 10 m/s, maximum acceleration = 6 m/s2
  • DMaximum velocity = 6 m/s, maximum acceleration = 0 m/s2
  • EMaximum velocity = 10 m/s, maximum acceleration = 2 m/s2

Q4:

At time 𝑑 seconds, a particle P has position vector βƒ‘π‘Ÿ relative to a fixed origin 𝑂, where βƒ‘π‘Ÿ=(2π‘‘βˆ’5)⃑𝑖+ο€Ήπ‘‘βˆ’9π‘‘ο…βƒ‘π‘—οŠ©, 𝑑β‰₯0.

Find the velocity of P when 𝑑=2.

  • A⃑𝑣=ο€Ί2βƒ‘π‘–βˆ’5⃑𝑗 m/s
  • B⃑𝑣=ο€Ί2⃑𝑖+3⃑𝑗 m/s
  • C⃑𝑣=ο€Ί2βƒ‘π‘–βˆ’10⃑𝑗 m/s
  • D⃑𝑣=ο€Ίβˆ’βƒ‘π‘–βˆ’10⃑𝑗 m/s
  • E⃑𝑣=ο€Ί3⃑𝑖+2⃑𝑗 m/s

Find the acceleration of P when 𝑑=2.

  • Aβƒ‘π‘Ž=2⃑𝑗 m/s2
  • Bβƒ‘π‘Ž=ο€Ί2⃑𝑖+3⃑𝑗 m/s2
  • Cβƒ‘π‘Ž=12⃑𝑖 m/s2
  • Dβƒ‘π‘Ž=2⃑𝑖 m/s2
  • Eβƒ‘π‘Ž=12⃑𝑗 m/s2

Q5:

At time 𝑑 seconds, a particle P has a position vector βƒ‘π‘Ÿ relative to a fixed origin 𝑂, where βƒ‘π‘Ÿ=2𝑑⃑𝑖+ο€Ή25π‘‘βˆ’2.5π‘‘ο…βƒ‘π‘—οŠ¨οŠ¨ and 𝑑β‰₯0.

Find the speed of P when 𝑑=2.

Find the acceleration of P and explain why it is constant.

  • Aβƒ‘π‘Ž=ο€Ί4⃑𝑖+25⃑𝑗/ms. The acceleration is constant because it is independent of 𝑑.
  • Bβƒ‘π‘Ž=ο€Ί4βƒ‘π‘–βˆ’5⃑𝑗/ms. The acceleration is constant because it is independent of 𝑑.
  • Cβƒ‘π‘Ž=ο€Ί4⃑𝑖+5⃑𝑗/ms. The acceleration is constant because it is independent of 𝑑.
  • Dβƒ‘π‘Ž=ο€Ί4βƒ‘π‘–βˆ’5𝑑⃑𝑗/ms. The acceleration is constant because only one component is dependent on 𝑑.
  • Eβƒ‘π‘Ž=ο€Ί4π‘‘βƒ‘π‘–βˆ’5⃑𝑗/ms. The acceleration is constant because only one component is dependent on 𝑑.

Find the magnitude of the acceleration, giving your answer in exact form.

  • A||βƒ‘π‘Ž||=41/ms
  • B||βƒ‘π‘Ž||=√89/ms
  • C||βƒ‘π‘Ž||=17/ms
  • D||βƒ‘π‘Ž||=172/ms
  • E||βƒ‘π‘Ž||=√41/ms

Q6:

A particle P moves on the π‘₯-axis. At time 𝑑 seconds, the velocity of P is 𝑣 m/s in the positive π‘₯-direction, where 𝑣 is given by 𝑣=3𝑑0≀𝑑≀4,βˆ’20+12π‘‘βˆ’π‘‘π‘‘>4.

Find the acceleration of P when 𝑑=2.

Find the acceleration of P when 𝑑=5.

Find the maximum velocity of P.

Q7:

A particle P of mass 0.25 kg is acted upon by a single force ⃑𝐹 N. Its position relative to a fixed origin 𝑂 at time 𝑑 seconds is βƒ‘π‘Ÿ m, where βƒ‘π‘Ÿ=2𝑑⃑𝑖+128π‘‘βƒ‘π‘—οŠ¨οŠ±οŽ οŽ‘, 𝑑β‰₯0.

Find the speed of P when 𝑑=4, giving your answer to 2 decimal places.

Find ⃑𝐹 when 𝑑=4.

  • Aβƒ‘π‘–βˆ’4⃑𝑗 N
  • B⃑𝑖+4⃑𝑗 N
  • C4+3⃑𝑗 N
  • Dβƒ‘π‘–βˆ’0.75⃑𝑗 N
  • E⃑𝑖+0.75⃑𝑗 N

Q8:

A particle of mass 500 grams moving on a plane is acted upon by a variable force ⃑𝐹 N. Its velocity at time 𝑑 seconds is given by ⃑𝑣=(4π‘‘βˆ’2)⃑𝑖+10𝑑⃑𝑗/ms, 𝑑β‰₯0.

Find ⃑𝐹 when 𝑑=5.

  • Aο€Ί4⃑𝑖+100⃑𝑗 N
  • Bο€Ί2⃑𝑖+50⃑𝑗 N
  • Cο€Ί18⃑𝑖+250⃑𝑗 N
  • Dο€Ί50⃑𝑖+2⃑𝑗 N
  • Eο€Ί9⃑𝑖+125⃑𝑗 N

Q9:

A particle P moves in a straight line so that, at time 𝑑 seconds, its displacement, 𝑠 m, from a fixed point 𝑂 is given by 𝑠=14𝑑0≀𝑑≀20,βˆšπ‘‘+5𝑑>20.

Find the velocity of P when 𝑑=5.

  • A32 m/s
  • B34 m/s
  • C5 m/s
  • D14 m/s
  • E54 m/s

Find the velocity of P when 𝑑=31.

  • A16 m/s
  • B3 m/s
  • C112 m/s
  • D314 m/s
  • E6 m/s

This lesson includes 81 additional question variations for subscribers.

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