The velocity (feet per second) of an object is given by the equation v = 3(2t + 1)0.4.
What is the equation for the acceleration of the object?
What is the equation for the displacement of the object? The displacement is initially 25 feet.
What are the values of the displacement, velocity and acceleration at time t = 2 seconds?
assuming the velocity is as written,
a = 3(2) = 6
If you meant 3(2t+1)^0.4, then
a = 2.4/(2t+1)^0.6
as usual,
s = 25 + 1/2 at^2
= 25 + 3t^2
or
= 25 + 1.2(2t+1)^1.4
just plug in your values.
Where did the 25 come from?
To find the equation for the acceleration of the object, we need to take the derivative of the velocity equation with respect to time. The acceleration is the rate at which the velocity is changing with time.
Given: v = 3(2t + 1)0.4
First, let's simplify the equation by distributing and combining like terms:
v = 3(0.4)(2t + 1)
v = 1.2(2t + 1)
v = 2.4t + 1.2
Now, let's take the derivative of v with respect to time (t):
a = d(v)/dt = d(2.4t + 1.2)/dt = 2.4
The equation for the acceleration of the object is simply a = 2.4.
To find the equation for the displacement of the object, we need to integrate the velocity equation with respect to time. Displacement is the accumulated change in position over time.
Given: v = 3(2t + 1)0.4
First, let's rewrite the equation:
v = 2.4t + 1.2
Next, we integrate v with respect to t to find the equation for displacement (s):
s = ∫(2.4t + 1.2) dt
s = (2.4/2)t^2 + 1.2t + C
Since the displacement is initially 25 feet, we can find the value of C using that information:
25 = (2.4/2)(0^2) + 1.2(0) + C
25 = C
Therefore, the equation for the displacement of the object is:
s = (2.4/2)t^2 + 1.2t + 25
At t = 2 seconds, we can substitute that value into the equations for displacement, velocity, and acceleration to find their respective values.
Displacement:
s = (2.4/2)(2^2) + 1.2(2) + 25
s = 2.4(4) + 1.2(2) + 25
s = 9.6 + 2.4 + 25
s = 37 feet
Velocity:
v = 2.4(2) + 1.2
v = 4.8 + 1.2
v = 6 ft/s
Acceleration:
a = 2.4
Therefore, at t = 2 seconds, the displacement of the object is 37 feet, the velocity is 6 ft/s, and the acceleration is 2.4 ft/s^2.