1. Consider a projectile, say a frictionless teddy bear, which is thrown at an angle of theta� with an initial
velocity of v0.
� a) For a fi�xed angle, �and the maximum height of the teddy bear.
� b) Calculate the distance that the teddy bear travels before returning (gently, of course) to the
ground (which is assumed to be
at).
� c) Use these pieces of information to �nd the angle which produces the maximum of the sum of height and distance.
usage of function. in these question.
Vi = Vo sin T
u = Vo cos T
v = Vi - gt
max height when v = 0
0 = Vosin T - g t
t = Vo sin T / g
h = 0 + Vi t - .5 g t^2
max h = Vo sin T [Vo sin T/g] - .5 g[Vo sin T / g]^2
max h = [.5 /g][VosinT]^2
(this is quicker to get using potential and kinetic energy argumenents. Note max height of course when T = 90 degrees, straight up)
Now part b
the total time in the air is twice the time needed to reach max altitude so
total t = 2 Vo sin T/g
d = u t = Vo t cos T
d = Vo [ 2 Vo sinT/g]cos T
d = [2 Vo^2/g] sin T cos T
part c
Sum = S = [.5 /g][VosinT]^2 + [2 Vo^2/g] sin T cos T
dS/dT = 0 at max
0= [.5Vo^2/g]sin Tcos T+[2Vo^2/g][-sin^2T+cos^2T]
0=2 cos^2T -.5 sin T cosT -2 sin^2T
but 2 (cos^2T-sin^2T)=2 cos2T
0= 2 cos 2T -.5 sinTcosT
but sinTcosT =.5 sin2T
so
0=2 cos2T -.25 sin2T
sin2T/cos2T = 8 = tan 2T
2T = 82.8 degrees
T = 41.4 degrees
Interesting, max range is at 45 degrees. I may have made an arithmetic error in that mess.
@Damon : Appreciate it sir. u are a life saver. i was on track and needed a kick . thanks alot.
To answer the given questions, you would need to use the equations of motion for projectile motion. The key equations are:
1. Vertical motion:
- Height (h) at any time (t): h = v0sin(theta)t - (1/2)gt^2
- Maximum height (hmax): hmax = (v0^2sin^2(theta))/(2g)
2. Horizontal motion:
- Distance traveled (d) at any time (t): d = v0cos(theta)t
- Time of flight (t_flight): t_flight = (2v0sin(theta))/g
- Maximum distance (dmax): dmax = v0^2sin(2theta)/g
Now let's use these equations to answer the specific questions:
a) To find the maximum height of the teddy bear for a fixed angle, you need to find the maximum point of the vertical motion. This occurs when the vertical velocity component becomes zero (at the top of the trajectory).
To find the time at which the vertical velocity becomes zero, you set v0sin(theta) - gt = 0 and solve for t. Once you have the time, substitute it back into the height equation h = v0sin(theta)t - (1/2)gt^2 to calculate the maximum height.
b) To calculate the distance traveled by the teddy bear before returning to the ground, you need the time of flight, which is the total time taken for the bear to reach the ground after being launched. Use the equation for horizontal motion to find the distance traveled by substituting the time of flight into the equation d = v0cos(theta)t_flight.
c) To find the angle that produces the maximum sum of height and distance, you can calculate the sum of the maximum height (hmax) and maximum distance (dmax) for different angles. Iterate through a range of angles and calculate the sum of hmax and dmax for each angle. The angle that gives the highest sum will be the angle that produces the maximum overall distance.
To implement these calculations, you can use a programming language like Python or any other mathematical tool that supports equations and numerical calculations.