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The independence of the horizontal and vertical components of the motion of the object is often used in these type of physics problems. It also accounts for the counterintuitive observation that a bullet fired horizontally from a gun takes the same time to fall to the ground as a bullet that is dropped from the same height. To be able to solve projectile problems you will often need to into its components. Use the up and down arrow keys on your keyboard to raise and lower the cannon barrel. The left and right arrow keys alter the amount of power in the explosion. Press the fire button to fire obviously. The path with air resistance can be switched on using the tick box and when you feel there are too many traces on the screen they can be cleared using the clear trace button. Use the cannon to find the maximum height, maximum range for a particular velocity of cannon ball. Compare the parabolic flight path of the cannonball with the path of the cannonball taking air resistance into account. How is it different? For a projectile without air resistance, the angle that gives the maximum range is 45° try and find the angle for maximum distance. Does the angle change with velocity? Balistics Consider a cannonball fired from a cannon. The barrel of the cannon points at an angle θ from the horizontal. This is known as its elevation. When the cannon is fired the explosion and expansion of the gases created accelerate the cannonball until it leaves the barrel. The velocity with which it leaves the barrel is known as the muzzle velocity. To understand the motion of the ball, the velocity can be resolved into horizontal and vertical components. Experiments, taking multiple images with strobe light of projectiles, show that the horizontal component of velocity is constant with time. That is, the object travels at the same horizontal speed throughout the motion. The vertical component has gravity acting against the projectile so its motion is that of an object that has been thrown into the air or dropped from a platform. When these two motions are combined, the resulting path is known as a parabola. Even for extended complicated objects the motion is roughly parabolic. Divers diving into a swimming pool, motorcross riders. Although the motion of the object may be very complicated if you work out where the centre of mass is and follow this point, it will move in a parabolic trajectory. Projectile motion problems can be solved using the. Maximum Height When the projectile reaches its maximum height its vertical component of velocity will be zero. We can use the linear equations of motion to calculate this height, h. The final velocity is zero. If we know the initial velocity, we can work out the time. The vertical motion looks like that of a ball being thrown upward. At its highest point the velocity will be zero. We just need to find the time that it takes and multiply by two, since what goes up, must come down. Maximum Range with Angle But what about the greatest range? It will have a maximum when sine is 1. Take the equation for range from a height, and differentiate. A rather tedious exercise in calculus lead us to the following expression. If the object is light, then it does not have the inertia to push through the air. Air resistance, often called drag creates an additional force on the projectile that acts in the opposite direction to the velocity. For objects that move through air and at high velocity, the air resistance is proportional to the square of the velocity. We know that gravity will always act vertically downward on the projectile. The air resistance creates a force, that is always in the opposite direction to the velocity. In the x direction it will act to decelerate the projectile reducing its range. It will also act on the vertical component of velocity slowing the projectile on both the upward and downward part of the motion. The motion is described by summing the forces in the x and y directions. Since acceleration is the second derivative of the position we can write a differential equation which the motion. Numerical solution is possible however. To solve numerically, we make the assumption that the over a small enough interval of time we can come approximate the solution by applying the linear equations of motion, to obtain velocities and then the position of the projectile. Before we do that however we should rewrite the differential equations. The new velocities will be.