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Question 1

A particle moves with velocity $v(t) = t^3 - 3t^2 + t + 2 \text{ m s}^{-1}$ . Find the total distance traveled between $t = 0$ and $t = 4 \text{ s}$ .

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Question 2

A particle has acceleration $a(t) = 6t - 2 \text{ m/s}^2$ , with $v(1) = 3 \text{ m/s}$ and $s(1) = 5 \text{ m}$ . Find the displacement function $s(t)$ and compute $s(4)$ .

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Question 3

A cyclist rides so that $v(t) = 5\sin t \text{ m s}^{-1}$ , with $s(0) = 0$ . Find the net displacement from $t = 0$ to $t = \pi$ s.

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Question 4

A particle moves along a straight line such that its acceleration $a$ at time $t$ seconds is given by $a(t) = e^{-t} \text{ m s}^{-2}$ . At $t = 0$ , the velocity of the particle is $v(0) = 5 \text{ m s}^{-1}$ .

1.

Find an expression for the velocity of the particle, $v(t)$ , at time $t$ .

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2.

Determine the total distance traveled by the particle as $t \to \infty$ .

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Question 5

Given velocity $v(t) = 2t^2 - t + 1 \text{ m/s}$ , find the acceleration $a(t)$ and the displacement between $t = 1 \text{ s}$ and $t = 3 \text{ s}$ .

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Question 6

A ball is thrown vertically upward with initial velocity $v_0 = 20 \text{ m s}^{-1}$ from ground level. Take $a(t) = -9.8 \text{ m s}^{-2}$ .

Find (a) the time to reach maximum height; (b) the maximum height reached.

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Question 7

A particle has acceleration $a(t) = 3t \text{ m\,s}^{-2}$ and initial velocity $v(0) = 4 \text{ m\,s}^{-1}$ .

Find $v(t)$ and the distance traveled from $t = 0$ to $t = 4$ seconds.

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Question 8

Given velocity $v(t) = t^2 - 4t + 3\text{ m/s}$ , find the total distance traveled from $t = 0$ to $t = 5\text{ s}$ .

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Question 9

The displacement of a particle is given by $s(t) = t^4 - 4t^3 + 6t^2 \text{ m}$ for $t \ge 0$ , where $t$ is measured in seconds.

Find its velocity $v(t)$ , acceleration $a(t)$ , and the times when the particle changes direction for $t \ge 0$ .

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Question 10

A car accelerates from rest with acceleration $a(t) = 9 - 0.5t \, \text{m s}^{-2}$ for $t \geq 0$ .

1.

Find expressions for the velocity $v(t)$ and displacement $s(t)$ in terms of $t$ .

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2.

Determine the time when the car comes momentarily to rest.

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Question 11

The question involves finding the velocity and displacement of a particle moving with constant acceleration using integration and initial conditions.

A particle moves with constant acceleration $a = 5 \text{ m s}^{-2}$ , initial velocity $v(0) = 2 \text{ m s}^{-1}$ , and initial displacement $s(0) = 0 \text{ m}$ .

Find expressions for $v(t)$ and $s(t)$ .

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Question 12

A particle moves with velocity $v(t) = t^3 - 6t^2 + 9t \text{ m s}^{-1}$ . Determine the total distance traveled between $t = 0$ and $t = 3 \text{ s}$ .

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Question Type 3: Integrating kinematic variables Exercises