Projectile Motion of Jumping Horses

    One way to analyze the physics of jumping horses is to consider their movement in terms of projectile motion.  In order to do this, two equations, the range equation and the height equation, come in very handy.  However, one stipulation to keep in mind is that these equations are only true if the projected object lands on the same level from which is leaves.  (This is important to note because in many equestrian jumping sports a horse does not land on the same plane from which it took off.  A good example of this is cross-country jumping, in which horses travel over a variety of terrains, and are asked to jump unusual obstacles that are sometimes placed on hills.)

To begin our discussion of projectile motion, let's first look at the two important equations:

Range equation:    R = (v_o²/g) (sin 2θ)                            

Height equation:    H = (v_o² sin²θ)/(2g)

For this analysis we are going to determine the initial velocity at which the horse must leave the ground in order to get over a jump of a certain height and width. 

Example Question:  With what initial velocity will a horse need to take off in order to jump a 1.5m high fence?

*Note:  The horse needs to take off from a distance as far away as the jump is high.  This means that the range will be approximately equal to four meters, two before the jump and two after the jump, with the width of the jump being negligible (not true in cases of more advanced jumping, but not something that we will investigate at this point.) *

Solution:

We will assume that the horse will leave the ground at an angle of approximately 45°.  Of course this angle would change depending on the different properties of the intended jump, but for our purposes here it works just fine.

H = (v_o² sin²θ)/(2g)

v_o= √((H*2g)/(sin²θ))

v_o= √((1.5m*2*9.8m/s)/(sin²45)

v_o= 7.7m/s

It is pretty impressive to think that an animal that large can launch itself into the air at a velocity of almost 8m/s!  Something that is interesting to keep in mind is that this initial velocity is the velocity at which the horse takes off from the ground, not the velocity of the horse as it approaches the jump.  The faster a horse goes toward a jump does not necessarily determine with how high the horse will jump.

 ☺Interesting Horsey info:  In most equine jumping sports, the time that it takes for a horse to complete a course of jumps is timed.  The horse with the fastest time wins!  However, riders have to be careful that their horses still maintain their style as they are in the air, because if a horse knocks down one of the loosely placed rails of the jump, then that counts against their score.

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