Assignment:
1. (a) If you throw a golf ball straight up at a velocity vy, how long does it take to come back down?
(b) If a golf ball moves horizontally with a velocity vx, how far does it go in time t?
(c) If you throw a golf ball with an initial vertical velocity vy and an initial horizontal velocity vx, how far away does it strike the ground, assuming level ground?
2. Estimate the initial vertical velocity of the piano in the video at https://www.youtube.com/ watch?v=hZxCEkGk6HI. Explain how you got your answer.
3. The steam-powered catapult on the USS Abraham Lincoln can accelerate an F-18 Hornet to a speed of 170 knots. The deck height of this carrier is h = 26 m. If the pilot forgets to start the engines (unlikely) and falls asleep during the launch (even more unlikely) the Hornet becomes nothing more than a streamlined projectile. How far does this projectile go before hitting the water?
4. You ride a unicycle in a straight line at constant speed v. The radius of the wheel is r.
(a) What is the instantaneous acceleration (magnitude and direction) of a point at the top of the wheel?
(b) What is the instantaneous acceleration (magnitude and direction) of a point at the bottom of the wheel?
5. A good sports car can manage a lateral acceleration of 1.1g's. How fast can this car go around a flat freeway onramp with a radius of 38 meters?
6. If the radius of a highway corner is increased by a factor of two, by how much is the maximum safe speed increased?
7. The range equation,
R = v2 sin(2θ)/g
is valid for negligible air resistance and flat horizontal ground. The factor of 2θ means that there are generally two possible angles which can reach any valid range. For example, projectiles launched with equal velocities at θ = 60 and at θ' = 30 will land at the same spot. The two projectiles will not take the same time, though. During the 1980's, the US Army was actually working on an artillery system that utilized this idea. The howitzer would fire two shells: the first at the higher angle, then some precise time interval Δt later the second at the lower angle. If they got Δt right, both shells would hit the target at the same time.1
Derive an equation for Δt. Your answer should depend on R, v,θ , and θ'. (Somewhat interestingly, it does not depend on g...)
8. You have designed an egg-launcher that launches eggs with a speed v = 4.0 m/s. The horizontal distance from the egg-launcher to a high vertical wall is R = 4.0 m. You can adjust the launch angle θ, but not v or R. At what angle should you launch the egg so as to maximize the height h at which it hits the wall?
9. If you launch a projectile o↵ the top of a cli↵, with initial height h, the range R at which the projectile strikes the ground will depend on the initial speed v, the launch angle θ, and the gravitational field g. The relationship between R, v, g, and θ in this case is not the one given in problem 7 because that equation was derived for the case in which the launch point and impact point were at the same elevation, which is not the case here.
(a) Derive an equation for R as a function of v, θ, g, and h.
(b) Check your answer by letting h go to zero: the result should simplify to the sameelevation answer.
10. The h-dependent range equation derived in part (a) of the previous problem makes it easy to figure out how far something will go for a given launch angle. But what if you want to know what launch angle to use to hit a target at a known range R? It's possible to invert the equation, but it's quite dicult algebraically. . . Instead, let's use a computer.
What we have is R = f(θ), assuming that v and h are constants. The usual way to solve things computationally is to rearrange this like so: F(θ) ≡ ƒ(θ)- R. Having done that, we can plot F(θ), and graphically determine the value of θ for which F(θ) = 0.
Using Python, write a function F(θ) that returns the value of f(θ) R. Plot F(θ) versus θ,and using the zoom tool on the resulting graph determine the value of θ for which F(θ) = 0.
Report your answers to at least two decimal places. Be sure to save your work; you'll need it next week. Use these values: R = 2.5 m, h = 1.2 m, v = 4.8 m/s. There may be more than one correct answer.