Photo by Jeff Ott

One of the first topics in a physics course is the motion of projectiles, such as a shotput. A shotput is a very ideal projectile to look at because it isn't affected much by the air it moves through. One can therefore ignore effects like air drag or aerodynamic lift in the case of a shotput.

Projectile Motion

You have perhaps heard that a shotput follows a parabolic trajectory. A parabola is a line that is given by the equation y(x) = a + bx + cx², where a,b,c are constants. Why does a shotput follow a parabolic arc, and not some other curve?

The answer lies in the nature of motion in the presence of gravity. And to look at that, let's first drop the shotput from the top of a wall of height h (making sure no one we like is standing below). If we record its fall with a video or film camera, we can write its height y as a function of time t. Say the wall has a height of 10 m (a very high wall); you'd get a table of y(t) such as second column below (units are put in brackets, like [s] for seconds and [m] for meters):

It takes a little over 1.4 s for the shotput to drop from a height of 10 m. Note that the shotput falls larger distances in equal time increments as time increases; in other words, its speed, the rate at which it drops, increases with time.

Suppose in addition to the film camera you had one of those guns the cops use to measure speed (maybe I'll show how those work in a later chapter). Then you'd get the speed measurements v(t) shown in the third column. Notice that the downward (negative) speed increases linearly: each 0.1 s interval adds -0.98 m/s to the speed. This is known as constant acceleration: the acceleration of an object due to gravity near the surface of the Earth is a constant 0.98 m/s per 0.1 s, or 9.8 m/s per s downward, written 9.8 m/s². This constant is usually called g, and it's an important number for projectile motion:

g = 9.8 m/s² = 32 ft/s².

If, instead of dropping the shotput from rest at the top of the wall, we threw it upward, then it would start with a positive vertical speed, travel upward, reach a maximum height and then fall back down. There is really no difference in this case, and the vertical speed still has -9.8 m/s added to it every second by gravity. This is exactly what happens with the shotput!

While a shotput is traveling horizontally downfield, it is actually undergoing the exact same vertical motion that it would have if just thrown straight up. The shotput toss is just a combination of constant horizontal motion and downward vertical acceleration. Here's what it looks like from the side (hit RELOAD to replay this GIF animation 10 times):

I've used numbers approximately matching the world record shotput trajectory of Randy Barnes, USA, 1990. I've assumed he launched the shotput from a height of 2 m at a 45-degree angle to the ground. I label the vertical position of the shotput with y and the horizontal position (distance downfield) with x.

Since the shotput travels an equal horizontal distance every 0.1 s, the horizontal velocity v_x is constant! Gravity has no effect on horizontal motion, so the shotput simply coasts with the initial horizontal velocity that Randy gave it. However, gravity does affect the vertical velocity v_y in the same way it did in the earlier example: v_y changes by -0.98 m/s in each 0.1 s interval.

The combination of constant horizontal velocity and constant downward acceleration results in a parabolic trajectory as seen above. You can vary the launch angle and initial speed and see the resulting trajectories with the Projectile Applet.

Equations

Here are the equations that govern projectile motion, in case you need them. The initial conditions are that the object starts at x₀ and y₀ with horizontal and vertical speeds v_x0 and v_y0.

• horizontal velocity: v_x = v_x0
• vertical velocity: v_y = v_y0 - gt
• horizontal position: x = x₀ + v_x0t
• vertical position: y = y₀ + v_y0t - ¹/₂gt²

If one eliminates t from the last two equations, one gets the parabolic equation for y(x).

Applets

• Projectile Applet

Summary

• Gravity near the Earth's surface accelerates an object downward with the constant value g = 9.8 m/s².
• Projectile motion is the combination of constant horizontal motion and downward acceleration due to gravity.
• The resulting trajectory is a parabola.