

Vector Problems
Name: Unknown
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1993
Question:
To any scientist it may concern: I am in desperate need of
physics help. I am having serious problems with my Physics 103: Mechanics
class. I have a problem with vectors in particular. For example: A particle
leaves an origin with an initial velocity of v=3.0i (i being the x component
of velocity vector v). It experiences a constant acceleration a=(1.0io.5j),
in m/s^2, (a) What is the velocity of the particle when it reaches its maximum
x coordinate? (b) Where is the particle at this time? A detailed
explanation of how to solve this problem and reasons for respective solution
steps would be greatly appreciated.
Replies:
Okay, the important thing is to write down the equation of motion
for each component, the x and the y, separately. The definition of
acceleration in the x direction is a_x = d2x / dt2 , or the second derivative
of x with respect to time. Since the acceleration is constant, if we
integrate this equation, we get dx/dt = a_x * t + (v_x)_0, where dx/dt is the
velocity of the x coordinate and (v_x)_0 is the initial velocity in the x
direction. If we integrate again, we get x(t) = 0.5* a_x * t^2 + (v_x)_0 * t
+ x_0, where x(t) is the x coordinate as a function of time and x_0 is the
initial position. Similarly, y(t) = 0.5* a_y * t^2 + (v_y)_0 * t + y_0. Now
we plug in the initial conditions. In the problem, we start at the origin, so
x_0 = y_0 = 0. The acceleration is (1.0i,0.5j) [the particle will be pulled
down and to the left] and the initial velocity is (3.0i, 0.0j). Substituting
these numbers into the two equations, we get x(t) = 0.5 * t^2 + 3.0 * t [ a_x
= 1.0,(v_x)_0 = 0.5, x_0 = 0] y(t) = 0.25* t^2 [ a_y = 0.5,(v_y)_0 =
0.0, y_0 = 0] Now we can answer the questions.
(a) What is the velocity of
the particle when it reaches its maximum x coordinate? Well, it should be
apparent that the velocity in the x coordinate will be zero when the particle
reaches its maximum value of x. But what is the y component of the velocity
at this time? Well, first we need to know at what TIME the particle is at its
maximum x value. To find this time, we differentiate x(t) and set the result
equal to zero; dx/dt =  t + 3.0 = 0 ; therefore, t=3.0 is the time at which
x(t) takes on it maximum value (this can be checked by plugging it into
d2x/dt2 and checking that the answer is negative, i.e., x(t) is "concave down"
at this critical point). Next, we differentiate y(t), which results in dy/dt
= 0.5* t. Plugging in t=3, we get dy/dt = 1.5, and so the velocity of the
particle when it reaches its maximum x coordinate is ... (0.0i, 1.5j) m/sec.
b) Where is the particle at this time? Just plug this value of t into the two
equation above, and get x(3) = 0.5 * 9 + 3.0 * 3 = 4.5 y(3) = 0.25 * 9 = 
2.25 and the position is (4.5i, 2.25j) meters.
Robert Topper
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Update: June 2012

