Solving projectile problems

Solving projectile problemsNow that we have the range equation (6.10), we can calculate how far a soccer ball goes when kicked upward at an angle of 30º with an initial velocity of 10 m/s. If we use the values v0 = 10 m/s and θ = 30º to plug into equation (6.10), then we find that the range of the ball is 8.8 m, which is approximately 30 ft. Read the text aloud
Looking at the equation we see that the speed appears as v2, which tells us that the range increases with the square of the speed. If you launch at twice the speed, the ball will go four times as far. Kicking the ball four times faster gets you a factor of 16 increase in range! At speeds much over 10 m/s, air resistance becomes impossible to ignore, so the range equation is not accurate at higher speeds! Read the text aloud Show Angular dependence
In the range equation (6.10), the range depends on the initial projection angle θ through the product of sin θ and cos θ. When you multiply sine and cosine together, the combined function is zero when either sine or cosine is zero. That means that the range is zero at 0º (when the ball is kicked along the ground) and at 90º (when kicked straight upward). In the absence of air resistance, the range is maximum at 45°—halfway between 0° and 90°. Read the text aloud
Solving a projectile problemNot all projectile problems start with an angle. Suppose you launch a marble horizontally off a cliff at a speed of 10 m/s. If the cliff is 10 m high, how far will the marble travel before it hits the ground?

The first step in the solution is to write down the equations of motion in the x- and y- directions, omitting any terms that are zero. In this problem, x0 = 0 and vy0 = 0. The marble starts at y0 = h with initial velocity vx0 in the x-direction. The y equation for position has only one unknown (time), so solve it for the time. The time is substituted back into the position equation in the x-direction to calculate the answer. The marble travels 14.2 m before it hits the ground. Read the text aloud
For an initial velocity of 10 m/s, tabulate the range for projection angles 30°, 40°, 45°, 50°, and 60°. What angle produces the maximum range? Show

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