The acceleration of a particle at time #t# seconds is given by # a = 6t#. Given that #v=10# when #t=0# and that #x=7# when #t=0# find #x# when #t=2#?
The position
# x = t^3 + 10t + 7 #
And so when
# x = 35#
This is a First Order separable Differential Equation which we can just integrate to get:
Again this is a First Order separable Differential Equation which we can just integrate to get:
By signing up, you agree to our Terms of Service and Privacy Policy
To find (x) when (t = 2), integrate the acceleration function (a = 6t) to get the velocity function (v(t)), then integrate the velocity function to get the displacement function (x(t)). Given that (v = 10) when (t = 0), you can find the constant of integration for velocity, and since (x = 7) when (t = 0), you can find the constant of integration for displacement. Finally, plug in (t = 2) into the displacement function to find (x).

Integrate (a = 6t) to find (v(t)): [v(t) = \int 6t dt = 3t^2 + C_1]

Use the given condition (v = 10) when (t = 0) to find (C_1): [10 = 3(0)^2 + C_1 \Rightarrow C_1 = 10]

Integrate (v(t) = 3t^2 + 10) to find (x(t)): [x(t) = \int (3t^2 + 10) dt = t^3 + 10t + C_2]

Use the given condition (x = 7) when (t = 0) to find (C_2): [7 = (0)^3 + 10(0) + C_2 \Rightarrow C_2 = 7]

Now, plug in (t = 2) into the displacement function to find (x): [x(2) = (2)^3 + 10(2) + 7 = 8 + 20 + 7 = 35]
So, (x = 35) when (t = 2).
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
 How do you find the volume bounded by #x^2y^2+16y^2=6# and the x & y axes, the line x=4 revolved about the xaxis?
 How do you find the general solution to #dy/dx+e^(x+y)=0#?
 What is the surface area of the solid created by revolving #f(x)=2x^33x^2+6x12# over #x in [2,3]# around the xaxis?
 How do you solve the differential #dy/dx=(x4)/sqrt(x^28x+1)#?
 How do you use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region #y=1/x# and #2x+2y=5# rotated about the #y=1/2#?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7