Active Figure 4.7 - The Projectile Path
Instructions: The simulation below illustrates the motion of a projectile launched at different speeds and angles.
Display in a New Window | Explore with this Active Figure Explore
A projectile is launched with a launch angle of 70° with respect to the horizontal direction and with initial speed 44 m/s.
(A) How do the vertical and horizontal components of its velocity vary with time?
(B) How long does it remain in flight?
(C) For a given launch speed, what launch angle produces the longest time of flight?
Conceptualize
Consider the projectile to be a point mass that starts with an initial velocity, upward and to the right, with forces from air resistance neglected. There is no force acting horizontally to accelerate its horizontal motion, while its vertical motion is accelerated downward by gravity. Therefore as the projectile moves to the right at a constant rate, the vertical part of its motion consists of first rising upward and then moving downward until the projectile strikes the ground. Use the simulation to display the projectile motion.
Categorize
The velocity has components in both the x and y directions, so we categorize this as a problem involving particle motion in two dimensions. The particle also has only a y component of acceleration, so we categorize it as a particle under constant acceleration in the y direction and constant velocity in the x direction.
(A) How do the vertical and horizontal components of its velocity vary with time?
Analyze The initial velocity in the x-direction vxi is related to the initial speed by
vxi = vi cos 70°
The constant velocity in the
x-direction means that the equation describing the time dependence of
x for the particle, with
x0 taken as 0, is
x = x0 + vxit = 0 +
m/s t
The equation for the vertical coordinate, which is constantly accelerating downward at
g = 9.8 m/s
2, is
y = y0 + vyit - ½gt2 = (
m/s) t + (
m/s2) t2
Finalize
The -½
gt2 term is negative. The other time-dependent term is proportional to
t and positive. Which of the two dominates at small
t ? Which term's magnitude gets larger faster as
t gets large? What effect does that have on the sign of the
y coordinate as
t starts out small and then gets larger? Is this consistent with the path you expect the projectile to take?
(B) How long does it remain in flight?
Analyze
The y-component of the projectile's velocity decreases by 9.8 m/s for each second of flight as the projectile rises. Therefore it takes a time of
for the vertical component of velocity to reach a value of 0, which occurs at the projectile's maximum height. At each height on the way down the particle has regained the same speed and has the same acceleration as it had on the way up, so that the complete time of flight is twice the time to reach the maximum height, and is equal to
In the present problem, that expression gives
tflight =
s
(C) For a given launch speed, what launch angle produces the longest time of flight?
Analyze
The time of flight for a given initial speed vi,
is largest when sin
θ is largest, which is at
θ =
°.
What does that correspond to physically?
Finalize
Is the result consistent with experience? Try the simulation for various launch angles and speeds, and check if the computer calculation shows this same result.