A 1.8-kg object moves in the x direction according to the following function:
x(t) = 2t2 + 3t − 5
(SI units). What is the force on the object after 2.7 s?
To calculate the force on the object after 2.7 seconds, you need to find the derivative of the position function with respect to time. The derivative will give you the velocity function, and from there you can calculate the force using Newton's second law.
Step 1: Find the derivative of the position function x(t) = 2t^2 + 3t - 5 with respect to time.
x'(t) = 4t + 3
Step 2: Evaluate the velocity function at t = 2.7 seconds.
x'(2.7) = 4(2.7) + 3
= 10.8 + 3
= 13.8 m/s
Step 3: Calculate the force using Newton's second law, which states that force (F) is equal to mass (m) multiplied by acceleration (a):
F = m * a
In this case, the mass of the object (m) is given as 1.8 kg. The acceleration (a) can be calculated as the derivative of the velocity function:
a = x''(t) = 4
Finally, substitute the values into the equation:
F = m * a
= 1.8 kg * 4
= 7.2 N
Therefore, the force on the object after 2.7 seconds is 7.2 Newtons.
To find the force on the object after 2.7 seconds, we need to calculate the second derivative of the position function and multiply it by the mass of the object. The second derivative represents acceleration, and multiplying it by mass gives us the force.
First, let's find the second derivative of the position function, x(t), with respect to time:
x(t) = 2t^2 + 3t - 5
To find the first derivative, we can differentiate each term of the function separately:
dx(t)/dt = d(2t^2 + 3t - 5)/dt
= 4t + 3
Now, let's find the second derivative by differentiating dx(t)/dt:
d^2x(t)/dt^2 = d(4t + 3)/dt
= 4
Therefore, the second derivative of the position function is 4.
Now, we have the acceleration, which is the second derivative of the position function. To find the force, we multiply the acceleration by the mass of the object:
Force = mass * acceleration
= 1.8 kg * 4
= 7.2 N (SI units)
So, the force on the object after 2.7 seconds is 7.2 Newtons.