As a newspaper delivery boy, Jason needs to know his projectile motion to throw a paper horizontally from a height of 1.40 m to a door that is 13.5 m away. What must the paper’s velocity be for it to reach the door?
h = ½ g t² ... t = √(2 h / g)
v = d / t = 13.5 m / t
To calculate the velocity needed for the paper to reach the door while being thrown horizontally, we can use the formulas of projectile motion.
Projectile motion consists of two independent motions: horizontal motion with constant velocity and vertical motion with constant acceleration due to gravity.
First, let's determine the time it takes for the paper to travel horizontally to the door. We can use the formula:
distance = velocity * time
In this case, the distance is 13.5 m, and the velocity in the horizontal direction is unknown (as it's what we're trying to find). Therefore:
13.5 m = velocity * time
Since the horizontal velocity remains constant, the time it takes to travel horizontally is the same as the time it takes to fall vertically from a height of 1.40 m. The vertical motion can be calculated using the formula:
distance = initial velocity * time + (1/2) * acceleration * time^2
In this case, the distance is 1.40 m, the initial vertical velocity is 0 m/s (since the paper is thrown horizontally), and the acceleration due to gravity is approximately 9.8 m/s^2. Therefore:
1.40 m = 0 * time + (1/2) * 9.8 m/s^2 * time^2
Simplifying this equation, we get:
1.40 m = (4.9 m/s^2) * time^2
Now, we have two equations involving time:
13.5 m = velocity * time
1.40 m = (4.9 m/s^2) * time^2
We can solve the second equation for time:
1.40 m = (4.9 m/s^2) * time^2
Divide both sides by 4.9 m/s^2:
0.2857 s^2 = time^2
Taking the square root of both sides, we get:
time = 0.5357 s
Now, we can substitute this value of time into the first equation to solve for the velocity:
13.5 m = velocity * 0.5357 s
Divide both sides by 0.5357 s:
velocity = 13.5 m / 0.5357 s
Calculating this expression, we find that the required horizontal velocity for the paper to reach the door is approximately 25.173 m/s.