Question

A sailboat that is coasting with its sails down has velocity v(t)= 100/2t+1metres per second, at time t. At what time(s) is a(t)=−0.5 m/ˢ²

Answer 1

To find when the acceleration a(t) is -0.5 m/s² for the sailboat with velocity function v(t) = 100/(2t+1), we take the derivative of v(t) to get a(t) and then solve for time t.

Step-by-step explanation:

The given velocity function of the sailboat is v(t) = 100/(2t+1) metres per second. To find the time(s) when the acceleration a(t) = -0.5 m/s², we need to take the first derivative of the velocity function v(t) to get the acceleration function a(t).

1. Begin by finding the derivative of the velocity function to determine the acceleration function:
   a(t) = dv/dt.
2. Solve the acceleration equation a(t) = -0.5 m/s² for t.
3. Use algebraic manipulation to isolate t and find the time(s) when the acceleration is -0.5 m/s².

Let's start by finding the derivative of the velocity function:
v(t) = 100/(2t+1)

a(t) = dv/dt = d/dt [100/(2t+1)]

Using the quotient rule, we would find that:
a(t) = -200/(2t+1)²

Now we set the acceleration function equal to -0.5 m/s²:
-200/(2t+1)² = -0.5

Solving for t gives us the times when the acceleration is -0.5 m/s².