Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of [tex]$19.6 \, \text{m/s}$[/tex] from a height of [tex]$24.5 \, \text{m}$[/tex] above the ground. The height (in meters) of the stone above the ground [tex]$t$[/tex] seconds after it is thrown is given by:

[tex]\[ s(t) = -4.9t^2 + 19.6t + 24.5 \][/tex]

a. Determine the velocity [tex]\( v \)[/tex] of the stone after [tex]\( t \)[/tex] seconds.

[tex]\[ v(t) = \square \][/tex]

b. When does the stone reach its highest point?

c. What is the height of the stone at the highest point?

d. When does the stone strike the ground?

e. With what velocity does the stone strike the ground?

f. On what intervals is the speed increasing?



Answer :

Let's solve each part step-by-step.

### a. Determine the velocity [tex]\( v \)[/tex] of the stone after [tex]\( t \)[/tex] seconds.
The height function given is:
[tex]\[ s(t) = -4.9t^2 + 19.6t + 24.5 \][/tex]

To find the velocity, we need to take the derivative of the height function [tex]\( s(t) \)[/tex] with respect to [tex]\( t \)[/tex]. The derivative of [tex]\( s(t) \)[/tex] is:
[tex]\[ v(t) = \frac{d}{dt} s(t) = \frac{d}{dt} \left( -4.9t^2 + 19.6t + 24.5 \right) = -9.8t + 19.6 \][/tex]
So, the object's velocity after [tex]\( t \)[/tex] seconds is
[tex]\[ v(t) = -9.8t + 19.6 \][/tex]

### b. When does the stone reach its highest point?
The stone reaches its highest point when the velocity [tex]\( v(t) \)[/tex] is zero. Setting [tex]\( v(t) = 0 \)[/tex], we get:
[tex]\[ 0 = -9.8t + 19.6 \][/tex]
Solving for [tex]\( t \)[/tex], we find:
[tex]\[ t = \frac{19.6}{9.8} = 2 \text{ seconds} \][/tex]

### c. What is the height of the stone at the highest point?
To determine the height at the highest point, we substitute [tex]\( t = 2 \)[/tex] seconds into the height function [tex]\( s(t) \)[/tex]:
[tex]\[ s(2) = -4.9(2)^2 + 19.6(2) + 24.5 \][/tex]
[tex]\[ s(2) = -4.9(4) + 39.2 + 24.5 \][/tex]
[tex]\[ s(2) = -19.6 + 39.2 + 24.5 \][/tex]
[tex]\[ s(2) = 44.1 \text{ meters} \][/tex]

### d. When does the stone strike the ground?
The stone strikes the ground when the height [tex]\( s(t) \)[/tex] is zero. We solve the equation
[tex]\[ 0 = -4.9t^2 + 19.6t + 24.5 \][/tex]

By solving this quadratic equation, we find that the time [tex]\( t \)[/tex] when the stone strikes the ground is:
[tex]\[ t = 5 \text{ seconds} \][/tex]

### e. With what velocity does the stone strike the ground?
To find the velocity when the stone strikes the ground, we substitute [tex]\( t = 5 \)[/tex] seconds into the velocity function [tex]\( v(t) \)[/tex]:
[tex]\[ v(5) = -9.8(5) + 19.6 \][/tex]
[tex]\[ v(5) = -49 + 19.6 \][/tex]
[tex]\[ v(5) = -29.4 \text{ meters per second} \][/tex]

### f. On what intervals is the speed increasing?
Speed increases when the velocity becomes more negative after reaching the highest point. The highest point occurs at [tex]\( t = 2 \)[/tex] seconds, and the stone strikes the ground at [tex]\( t = 5 \)[/tex] seconds. Therefore, the interval during which the speed is increasing is
[tex]\[ (2, 5) \][/tex]

So the answers are as follows:

a. The object's velocity after [tex]\( t \)[/tex] seconds is [tex]\( v(t) = -9.8t + 19.6 \)[/tex]

b. The stone reaches its highest point at [tex]\( t = 2 \)[/tex] seconds.

c. The height of the stone at the highest point is [tex]\( 44.1 \)[/tex] meters.

d. The stone strikes the ground at [tex]\( t = 5 \)[/tex] seconds.

e. The stone strikes the ground with a velocity of [tex]\( -29.4 \)[/tex] meters per second.

f. The speed is increasing on the interval [tex]\( (2, 5) \)[/tex].