Given that the position function of a particle in motion is s(t)=cost-16t^2 where t is measured in seconds and s(t) is measured in feet find the particle's initial acceleration
How is it -33 ft/s
s(t)=cost-16t^2
v(t) = s'(t) = -sint-32t
a(t) = v'(t) = -cost-32
now plug in t=0
To find the particle's initial acceleration, we need to take the derivative of the position function with respect to time.
Given that the position function is s(t) = cos(t) - 16t^2, we can find the velocity function by taking the derivative:
v(t) = d/dt (s(t))
v(t) = d/dt (cos(t) - 16t^2)
The derivative of cos(t) is -sin(t), and the derivative of -16t^2 is -32t. So, we have:
v(t) = -sin(t) - 32t
Next, to find the particle's initial acceleration, we need to take the derivative of the velocity function.
a(t) = d/dt (v(t))
a(t) = d/dt (-sin(t) - 32t)
The derivative of -sin(t) is -cos(t), and the derivative of -32t is -32. So, we have:
a(t) = -cos(t) - 32
To find the particle's initial acceleration, we evaluate the acceleration function at t = 0.
a(0) = -cos(0) - 32
a(0) = -1 - 32
a(0) = -33 ft/s
Therefore, the particle's initial acceleration is -33 ft/s.
To find the particle's initial acceleration, we need to differentiate the position function with respect to time, t, to obtain the velocity function. Then, we differentiate the velocity function to obtain the acceleration function.
Given that the position function is:
s(t) = cost - 16t^2
We will first differentiate s(t) to find the velocity function, V(t):
V(t) = d/dt (cost - 16t^2)
To differentiate the position function with respect to time:
The derivative of cost is -sint, and the derivative of -16t^2 is -32t.
Therefore, the velocity function is:
V(t) = -sint - 32t
Next, we differentiate V(t) to find the acceleration function, A(t):
A(t) = d/dt (-sint - 32t)
To differentiate the velocity function with respect to time:
The derivative of -sint is -cost, and the derivative of -32t is -32.
Therefore, the acceleration function is:
A(t) = -cost - 32
To find the particle's initial acceleration, we need to evaluate the acceleration function at the initial time, t=0.
Setting t=0 in the acceleration function:
A(0) = -cos(0) - 32
The cosine of 0 is 1, so the initial acceleration is:
A(0) = -1 - 32
A(0) = -33 ft/s
Therefore, the particle's initial acceleration is -33 ft/s.