it is just a matter of integration and using initial conditions since in general dv/dt = a it implies v = integral a dt
v(t)_x = integral a_{x}(t) dt = alpha t^3/3 + c the integration constant c can be found out since we know v(t)_x at t =0 is v_{0x} so substitute this in the equation to get v(t)_x = alpha t^3 / 3 + v_{0x}
similarly v(t)_y = integral a_{y}(t) dt = integral beta - gamma t dt = beta t - gamma t^2 / 2 + c this constant c use at t = 0 v(t)_y = v_{0y} v(t)_y = beta t - gamma t^2 / 2 + v_{0y}
so the velocity vector as a function of time vec{v}(t) in terms of components as[ alpha t^3 / 3 + v_{0x} , beta t - gamma t^2 / 2 + v_{0y} ]
similarly you should integrate to find position vector since dr/dt = v r = integral of v dt
r(t)_x = alpha t^4 / 12 + + v_{0x}t + c let us assume the initial position vector is at origin so x and y initial position vector is zero and hence c = 0 in both cases
r(t)_y = beta t^2/2 - gamma t^3/6 + v_{0y} t + c here c = 0 since it is at 0 when t = 0 we assume
r(t)_vec = [ r(t)_x , r(t)_y ] = [ alpha t^4 / 12 + + v_{0x}t , beta t^2/2 - gamma t^3/6 + v_{0y} t ]
Answer: 2.80 N/C
Explanation: In order to calculate the electric firld inside the solid cylinder
non conductor we have to use the Gaussian law,
∫E.ds=Q inside/ε0
E*2πrL=ρ Volume of the Gaussian surface/ε0
E*2πrL= a*r^2 π* r^2* L/ε0
E=a*r^3/(2*ε0)
E=6.2 * (0.002)^3/ (2*8.85*10^-12)= 2.80 N/C
Answer:
The magnitude of the centripetal force to make the turn is 3,840 N.
Explanation:
Given;
radius of the cured road, r = 400 m
speed of the car, v = 32 m/s
mass of the car, m = 1500 kg
The magnitude of the centripetal force to make the turn is given as;
where;
Fc is the centripetal force
Therefore, the magnitude of the centripetal force to make the turn is 3,840 N.
Answer:
34.45m
Explanation:
Magnitude of a vector is equal to the square root of sum of squares of x & y vectors.
Magnitude = 
= 
=34.45m
Given Information:
slope angle = θ = 30°
spring constant = k = 30 N/m
compressed length = x = 10 cm = 0.10 m
mass of ice cube = m = 63 g = 0.063 kg
Required Information:
distance traveled by ice cube = d = ?
Answer:
distance traveled by ice cube = 0.48 m
Explanation:
Using the the principle of conversation of energy, the following relation holds true for this case,
mgh = 1/2*kx²
h = 1/2*kx²/mg
Where h is the height of the slope, m is the mass of ice cube, k is the spring constant and x is the compressed length o the spring and g is gravitational acceleration.
h = 1/2*kx²/mg
h = 1/2*30(0.1)²/0.063*9.8
h = 0.242 m
From trigonometry ratio,
sinθ = h/d
d = h/sinθ
d = 0.242/sin(30)
d = 0.48 m
Therefore, when the ice cube is released, it will travel a total distance 0.48 up the slope before reversing direction.
