Let s(t)=-5t^2+40t be the position equation for a rock thrown upwards. At what time t, in seconds, will the rock change directions?
at its vertex.
the x of the vertex is -b/(2a) = -40/-10 = 4
it will change direction at t = 4 seconds
Well, it seems like the rock is going to have a "change of heart" and change directions at some point. Let's find out when the rock decides to do a 180!
To determine when the rock changes directions, we can find the time t when the velocity is equal to zero. The velocity is the derivative of the position function, s(t).
To find the velocity, we take the derivative of s(t). Applying the power rule of derivatives, we have:
s'(t) = -10t + 40
Now, to determine when the velocity is equal to zero, we set s'(t) equal to zero and solve for t:
-10t + 40 = 0
Solving this equation, we find:
-10t = -40
t = (-40) / (-10)
t = 4
So, after juggling the numbers, we find that the rock changes directions at t = 4 seconds.
Keep in mind that this is a simplified model, so in real life, the rock might have some fancy tricks up its sleeve before deciding to change directions.
To determine the time at which the rock changes directions, we need to find when the velocity of the rock becomes zero.
The velocity equation is the derivative of the position equation:
v(t) = s'(t) = -10t + 40
Setting the velocity equation equal to zero and solving for t:
-10t + 40 = 0
Adding 10t to both sides:
40 = 10t
Dividing both sides by 10:
t = 4
Therefore, the rock will change directions at t = 4 seconds.
To find the time at which the rock changes directions, we need to identify when the velocity of the rock changes from positive to negative or vice versa.
The velocity function, v(t), is the derivative of the position function, s(t). So, let's differentiate s(t) to get v(t):
s(t) = -5t^2 + 40t
To differentiate s(t), we apply the power rule:
s'(t) = (-5) * 2t^(2-1) + 40 * 1t^(1-1)
= -10t + 40
Now, we need to find when the velocity, v(t) = -10t + 40, changes signs. In other words, we need to solve the equation:
-10t + 40 = 0
To solve for t, we isolate the variable:
-10t = -40
Dividing both sides of the equation by -10, we find:
t = 4
Therefore, the rock changes directions at t = 4 seconds.