an object is projected at an angle of elevation of 45 degree with a velocity of 100m/s.calculate its range and time of flight
up for time t then down for time t (parabola)
u = 100 cos 45 forever
Vi = 100 sin 45 at start
v = Vi - g t
at top v = 0
so
at top t = Vi/g = 100 sin 45 / 9.81
in the air for time = 2 t
so
range = 2 t u = 2 t (100 cos 45)
Time of flight formula
=2×100/9.8
=20.40s
Range formula
=100×100÷9.8
=1020.4
=10.2m
To calculate the range and time of flight of an object projected at an angle of elevation, we can use the following equations of motion:
1. Horizontal Range (R) = [(v^2 * sin(2θ)] / g
2. Time of Flight (T) = (2 * v * sin(θ)) / g
Where:
- v is the initial velocity (100 m/s in this case)
- θ is the angle of elevation (45 degrees in this case)
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
Let's calculate the range and time of flight step by step:
1. Calculate the value of sin(2θ):
- sin(2θ) = sin(2 * 45) = sin(90) = 1
2. Calculate the value of sin(θ):
- sin(θ) = sin(45) = √2/2 ≈ 0.707
3. Calculate the horizontal range (R):
- R = [(v^2 * sin(2θ)] / g
= [(100^2 * 1)] / 9.8 ≈ 1020.41 meters
4. Calculate the time of flight (T):
- T = (2 * v * sin(θ)) / g
= (2 * 100 * 0.707) / 9.8 ≈ 14.43 seconds
Therefore, the range of the object is approximately 1020.41 meters, and the time of flight is approximately 14.43 seconds.