Math 160  ·  Calculus

Rectilinear Motion

Position, velocity, acceleration & rates of change along a line

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1
Polynomial Motion Solution

The position function (in meters) of a particle moving along a horizontal line in \(t\) seconds is given by

\[ s = f(t) = 2t^3 - 3t^2 - 12t + 8, \quad t \geq 0 \]
(a)
Find the velocity and acceleration functions.
(b)
At what time(s) is the particle at rest?
(c)
On what time interval(s) is the particle moving to the right or left?
(d)
Determine the time interval(s) when the particle is speeding up or slowing down.
2
Projectile Motion Solution

If a ball is thrown vertically upward with a velocity of 80 ft/sec, its height after \(t\) seconds is

\[ s(t) = 80t - 16t^2 \]
(a)
What is the velocity of the ball after 1 sec?
(b)
What is the maximum height reached by the ball?
(c)
What is the velocity of the ball when it is 96 feet above the ground on its way up?
(d)
With what velocity does the ball hit the ground?
3
Law of Motion Solution

A particle moves according to a law of motion \(s(t) = t^3 - 12t^2 + 36t\), where \(t\) is measured in seconds and \(s\) in feet.

(a)
What is the velocity of the particle after 3 sec?
(b)
When is the particle at rest?
(c)
When is the particle moving in the positive direction?
(d)
Find the total distance traveled during the first 8 seconds.
(e)
What is the acceleration after 2 sec?
4
Projectile from Height Solution

A slingshot launches a stone vertically from the top of a wall. Its height (in feet) after \(t\) seconds is given by

\[ s(t) = 100 + 112t - 16t^2 \]
(a)
Find the velocity of the stone after 2 seconds.
(b)
Find the maximum height of the stone.
(c)
Find the velocity of the stone after it has risen 260 feet.
5
Horizontal Motion Solution

The position of an object moving horizontally after \(t\) seconds is given by \(s = f(t)\), where \(s\) is in feet and \(s \geq 0\) corresponds to positions right of the origin.

\[ f(t) = 24t - 4t^2, \quad 0 \leq t \leq 8 \]
(a)
Find the velocity function. When is the object stationary, moving right, and moving left?
(b)
Determine the velocity and acceleration at \(t = 1\).
(c)
Determine the acceleration when the velocity is zero.
(d)
On what intervals is the speed increasing?
6
Velocity Graph Analysis

The graph of the velocity function of a particle is shown below, where \(t\) is measured in seconds.
When is the particle speeding up? When is it slowing down?

Graph showing velocity v(t). It begins at (0,4), decreases to (1,2), rises slightly to (2,3), then decreases steadily, crossing the t-axis at (4,0) to end at (5,-5).