Worksheet: Linear Motion with Integration

In this worksheet, we will practice using integration to get the average and instantaneous velocities and displacement vectors of a particle in straight-line motion.

Q1:

If the acceleration of an object is given by aij(𝑑)=3βˆ’4𝑑, find the object’s velocity function given that the initial velocity is vi(0)=2.

  • Aβˆ’4j
  • B(3𝑑+2)βˆ’2𝑑ij
  • C(3𝑑+2)βˆ’8𝑑ij
  • D3π‘‘βˆ’2𝑑ij
  • E(3π‘‘βˆ’2)βˆ’2𝑑ij

Q2:

An object’s acceleration is given by aij(𝑑)=4(2𝑑)+6(2𝑑)cossin. Find the object’s velocity function if its initial velocity is v(0)=0.

  • A2(2𝑑)βˆ’(3(2𝑑)βˆ’3)sincosij
  • Bβˆ’2(2𝑑)βˆ’(3(2𝑑)βˆ’3)sincosij
  • Cβˆ’2(2𝑑)+(3(2𝑑)βˆ’3)sincosij
  • Dβˆ’4(2𝑑)+(6(2𝑑)βˆ’3)sincosij
  • Eβˆ’2(2𝑑)+3(2𝑑)sincosij

Q3:

If the acceleration of an object is given by aijk(𝑑)=2𝑑+1+π‘’βˆ’(9𝑑)sin, find the object’s position function given that the initial velocity is v(0)=0.

  • A2|𝑑+1|+12ο€Ήπ‘’βˆ’1ο…βˆ’19((9𝑑)βˆ’1)lncosijk
  • B2|𝑑+1|+12ο€Ήπ‘’βˆ’1+19((9𝑑)βˆ’1)lncosijk
  • C2|𝑑+1|+𝑒2+(9𝑑)9lncosijk
  • D2|𝑑+1|+𝑒2βˆ’(9𝑑)9lncosijk
  • E2|𝑑+1|+12𝑒+1+19((9𝑑)+1)lncosijk

Q4:

If the acceleration of an object is given by aij(𝑑)=(2+8𝑑)+(4π‘‘βˆ’2), find the object’s velocity function, given that the initial velocity is vi(0)=8.

  • Aο€Ή2𝑑+8𝑑+ο€Ή4π‘‘βˆ’2π‘‘ο…οŠ¨οŠ¨ij
  • Bο€Ή2𝑑+4𝑑+ο€Ή2π‘‘βˆ’2π‘‘ο…οŠ¨οŠ¨ij
  • Cο€Ή2𝑑+4π‘‘βˆ’8+ο€Ή2π‘‘βˆ’2π‘‘ο…οŠ¨οŠ¨ij
  • Dο€Ή2𝑑+4𝑑+8+ο€Ή2π‘‘βˆ’2π‘‘ο…οŠ¨οŠ¨ij
  • Eο€Ή2𝑑+8𝑑+8+ο€Ή4π‘‘βˆ’2π‘‘ο…οŠ¨οŠ¨ij

Q5:

If the velocity of an object is given by vij(𝑑)=(2𝑑)+(5𝑑)sincos, find the object’s position function given that the initial position is r(0)=0.

  • A2(1βˆ’(2𝑑))+5(5𝑑)cossinij
  • Bβˆ’12(1+(2𝑑))+(5𝑑)5cossinij
  • C(1βˆ’(2𝑑))+(5𝑑)cossinij
  • D12(1+(2𝑑))βˆ’(5𝑑)5cossinij
  • E12(1βˆ’(2𝑑))+(5𝑑)5cossinij

Q6:

If the acceleration of an object is given by aijk(𝑑)=+22βˆ’12𝑑, find the function describing the object’s position given that its initial position and velocity are both zero.

  • A𝑑2+11π‘‘βˆ’2π‘‘οŠ¨οŠ¨οŠ¨ijk
  • Bijk+22π‘‘βˆ’11π‘‘οŠ¨
  • C𝑑2+11π‘‘βˆ’2π‘‘οŠ¨οŠ¨οŠ©ijk
  • Dβˆ’12k
  • E𝑑+22π‘‘βˆ’11𝑑ijk

Q7:

The velocity of an object is given by vijk(𝑑)=12𝑑+9𝑑+βˆšπ‘‘οŠ¨. Find the function r(𝑑) describing the object’s position given that rij(1)=6+3.

  • A6𝑑+3𝑑+23βˆšπ‘‘οŠ¨οŠ©οŠ©ijk
  • B6𝑑+3𝑑+ο€Ό23βˆšπ‘‘βˆ’23ijk
  • C6+3𝑑+ο€Ώ12βˆšπ‘‘βˆ’12ijk
  • D6𝑑+3𝑑+12βˆšπ‘‘ijk
  • E6𝑑+3𝑑+ο€Ό23βˆšπ‘‘βˆ’23ijk

Q8:

If the acceleration of an object is given by aij(𝑑)=14+3π‘’οŠ±ο, find the function describing the object’s position given that its initial position is 9i and initial velocity βˆ’3j.

  • Aο€Ή7𝑑+9+ο€Ή3π‘’βˆ’3ο…οŠ¨οŠ±οij
  • Bο€Ή7𝑑+9+3π‘’οŠ¨οŠ±οij
  • C(14𝑑+9)+3𝑒ij
  • Dο€Ή7π‘‘βˆ’9+3π‘’οŠ¨οŠ±οij
  • Eο€Ή7𝑑+9ο…βˆ’ο€Ή3π‘’βˆ’3ο…οŠ¨οŠ±οij

Q9:

An object’s acceleration is given by aij(𝑑)=2𝑑+1+π‘’οŠ©ο. The object’s initial velocity is v(0)=0. Find the object’s velocity function.

  • Aln(|𝑑+1|)+13𝑒+1ij
  • B2(|𝑑+1|)+13ο€Ήπ‘’βˆ’1lnij
  • C2(|𝑑+1|)+𝑒3lnij
  • D2(|𝑑+1|)+13𝑒+1lnij
  • Eln(|𝑑+1|)+13ο€Ήπ‘’βˆ’1ij

Q10:

If the acceleration of an object is given by aijk(𝑑)=(2+8𝑑)+8+7, find the function describing the object’s velocity function given that its initial velocity is zero.

  • Aο€Ή2𝑑+4𝑑+8+7ijk
  • Bο€Ή2𝑑+8𝑑+8𝑑+7π‘‘οŠ¨ijk
  • C8i
  • D(2𝑑+4𝑑)+8𝑑+7𝑑ijk
  • Eο€Ή2𝑑+4𝑑+8𝑑+7π‘‘οŠ¨ijk

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