Question

A rocket is fired from rest at x = 0 and travels along a parabolic trajectory described by y 2 = [120(10 3 )x] m. If the x component of acceleration is a x = ( 4 1 t 2 ) , m/s 2 , where t is in seconds, determine the magnitude of the rockets velocity and acceleration when t = 10 s.

Answer 1

Answer:

The velocity is 1003.5 m/s  and the acceleration is 103.1 m/$s^{2}$.

Explanation:

We need to find the parameter equation of x. To find it, we will need to integrate the x-component acceleration equation given to us, twice. Acceleration is:

a = $\frac{dv}{dt}$

dv = adt

∫dv = ∫adt

$\int\limits^v_0 \, dv$  =   $\int\limits^t_0 {\frac{1}{4}t^{2} } \, dt$

v = $\frac{1}{12} t^{3} m/s$.

Velocity is:

v = $\frac{dx}{dt}$

dx=vdt

Again, integrate both sides:

∫dx = ∫vdt

$\int\limits^x_0 dx$ = $\int\limits^t_0 {\frac{1}{12} t^{3} } \, dt$

x = $\frac{1}{48} t^{4} m$

Substitute our x equation into our parameter equation of y.

y$^{2}$ = [120($10^{3}$)x] m

y$^{2}$ = [120($10^{3}$)($\frac{1}{48}$)($t^{4}$) ]

(take the square root of both sides and simplify)

y = 50$t^{2}$

Now that we can represent our equation with respect to time, we can take the derivative to figure out the velocity. Remember that taking the derivative of a position function gives us the velocity function.

y = 50$t^{2}$

vy= y = 100t

Let us write down the two equations for velocity we found:

vx = $\frac{1}{12} t^{3} m/s$

vy = 100t m/s

At t = 10 s:

vx = $\frac{1}{12} 10^{3}$ = 83.3 m/s

vy  = 100(10)=1000 m/s

The magnitude of velocity is:

v = $\sqrt{(vx)^{2}+(vy)^{2} }$

v = $\sqrt{(83.3)^{2}+(1000)^{2} } = 1003.5 m/s$

To figure out the acceleration, we need to figure out ay which can be found by taking the derivative of the vy equation,

vy = 100t m/s

ay = vy = 100 m/$s^{2}$

Since ax is given to us in the question, we have the following:

ax = $(\frac{1}{4}t^{2}) m/s^{2}$

ay = 100 m/$s^{2}$

At t = 10 s:

ax = $(\frac{1}{4}10^{2})} =25 m/s^{2$

ay = 100 m/$s^{2}$

The magnitude of acceleration is equal to:

a = $\sqrt{(ax)^{2}+(ay)^{2} }$

a = $\sqrt{(25)^{2}+(100)^{2} }$

a= 103.1 m/$s^{2}$

Final Answers:

v = 1003.5 m/s

a= 103.1 m/$s^{2}$