Throwing or shooting a projectile follows a parabolic course. If you know the initial velocity and angle of elevation of the projectile, you can find its time aloft, maximum height or range. You can also its altitude and distance travelled if given a time. This example problem shows how to do all of these.
Projectile Motion Example Problem:
A cannon is fired with muzzle velocity of 150 m/s at an angle of elevation = 45°. Gravity = 9.8 m/s2.
a) What is the maximum height the projectile reaches?
b) What is the total time aloft?
c) How far away did the projectile land? (Range)
d) Where is the projectile at 10 seconds after firing?
Let’s set up what we know. First, let’s define our variables.
V0 = initial velocity = muzzle velocity = 150 m/s
vx = horizontal velocity component
vy = vertical velocity component
θ = angle of elevation = 45°
h = maximum height
R = range
x = horizontal position at t=10 s
y = vertical position at t=10 s
m = mass of projectile
g = acceleration due to gravity = 9.8 m/s2
Part a) Find h.
The formulas we will be using are:
d = v0t + ½at2
vf – v0 = at
In order to find the distance h, we need to know two things: the velocity at h and the amount of time it takes to get there. The first is easy. The vertical component of the velocity is equal to zero at point h. This is the point where the upward motion is stopped and the projectile begins to fall back to Earth.
The initial vertical velocity is
v0y = v0·sinθ
v0y = 150 m/s · sin(45°)
v0y = 106.1 m/s
Now we know the beginning and final velocity. The next thing we need is the acceleration.
The only force acting on the projectile is the force of gravity. Gravity has a magnitude of g and a direction in the negative y direction.
F = ma = -mg
solve for a
a = -g
Now we have enough information to find the time. We know the initial vertical velocity (V0y) and the final vertical velocity at h (vhy = 0)
vhy – v0y = at
0 – v0y = -9.8 m/s2·t
0 – 106.1 m/s = -9.8 m/s2·t
Solve for t
t = 10.8 s
Now solve the first equation for h
h = v0yt + ½at2
h = (106.1 m/s)(10.8 s) + ½(-9.8 m/s2)(10.8 s)2
h = 1145.9 m – 571.5 m
h = 574.4 m
The highest height the projectile reaches is 574.4 meters.
Part b: Find total time aloft.
We’ve already done most of the work to get this part of the question if you stop to think. The projectile’s trip can be broken into two parts: going up and coming down.
ttotal = tup + tdown
The same acceleration force acts on the projectile in both directions. The time down takes the same amount of time it took to go up.
tup = tdown
ttotal = 2 tup
we found tup in Part a of the problem: 10.8 seconds
ttotal = 2 (10.8 s)
ttotal = 21.6 s
The total time aloft for the projectile is 21.6 seconds.
Part c: Find range R
To find the range, we need to know the initial velocity in the x direction.
v0x = v0cosθ
v0x = 150 m/s·cos(45)
v0x = 106.1 m/s
To find the range R, use the equation:
R = v0xt + ½at2
There is no force acting along the x-axis. This means the acceleration in the x-direction is zero. The equation of motion is reduced to:
R = v0xt + ½(0)t2
R = v0xt
The range is the point where the projectile strikes the ground which happens at the time we found in Part b of the problem.
R = 106.1 m/s · 21.6s
R = 2291.8 m
The projectile landed 2291.8 meters from the canon.
Part d: Find the position at t = 10 seconds.
The position has two components: horizontal and vertical position. The horizontal position, x, is far downrange the projectile is after firing and the vertical component is the current altitude, y, of the projectile.
To find these positions, we will use the same equation:
d = v0t + ½at2
First, let’s do the horizontal position. There is no acceleration in the horizontal direction so the second half of the equation is zero, just like in Part c.
x = v0xt
We are given t = 10 seconds. V0x was calculated in Part c of the problem.
x = 106.1 m/s · 10 s
x = 1061 m
Now do the same thing for the vertical position.
y = v0yt + ½at2
We saw in Part b that v0y = 109.6 m/s and a = -g = -9.8 m/s2. At t = 10 s:
y = 106.1 m/s · 10 s + ½(-9.8 m/s2)(10 s)2
y = 1061 – 490 m
y = 571 m
At t=10 seconds, the projectile is at (1061 m, 571 m) or 1061 m downrange and at an altitude of 571 meters.
If you need to know the velocity of the projectile at a specific time, you can use the formula
v – v0 = at
and solve for v. Just remember velocity is a vector and will have both x and y components.
This specific example can be easily adapted for any initial velocity and any angle of elevation. If the cannon is fired on another planet with a different force of gravity, just change the value of g accordingly.