A ball thrown with a speed of 100m/s attains a height of 150m (take g=9.8 per second square). Calculate:
1. Time of flight
2. Angle of projection
3. Range
To find the time of flight, angle of projection, and range of the ball, we can use the equations of motion for projectile motion. Projectile motion refers to the motion of an object that is launched into the air and moves along a curved path under the influence of gravity.
1. Time of flight (T):
The time of flight is the total time the ball spends in the air. Since the ball is thrown upward and then falls back down to the ground, the time of flight can be calculated as the sum of the time taken for the upward journey (going up) and the time taken for the downward journey (coming down).
- First, let's find the time taken for the upward journey. We need to use the equation:
v = u + gt
where:
v = final velocity (0 m/s at the peak, as the ball will momentarily stop at the highest point)
u = initial velocity (100 m/s)
g = acceleration due to gravity (-9.8 m/s^2, as it acts in the opposite direction to the initial velocity)
t = time
Rearranging the equation:
0 = 100 - 9.8t
Solving for t:
9.8t = 100
t = 100 / 9.8
- Next, let's find the time taken for the downward journey. The time taken for the downward journey is the same as the time taken for the upward journey, as the ball is subject to symmetrical motion. Therefore, the time taken for the downward journey is also 100 / 9.8.
Adding the times for the upward and downward journeys, the total time of flight is:
T = 2 * (100 / 9.8)
2. Angle of projection (θ):
The angle of projection is the angle between the horizontal direction and the initial velocity vector of the ball. We can find this angle using the trigonometric relationships between the components of the initial velocity (horizontal and vertical).
- Let's consider the horizontal component of the initial velocity (u_x) and the vertical component of the initial velocity (u_y). Since the ball is launched at an angle, we can find these components using the following relationships:
u_x = u * cos(θ)
u_y = u * sin(θ)
where:
u = initial velocity (100 m/s)
θ = angle of projection (to be determined)
cos = cosine function
sin = sine function
Dividing the equation u_y = u * sin(θ) by the equation u_x = u * cos(θ), we get:
tan(θ) = u_y / u_x
Substituting the values:
tan(θ) = u * sin(θ) / u * cos(θ)
tan(θ) = sin(θ) / cos(θ)
Taking the inverse tangent of both sides to isolate θ:
θ = arctan(u_y / u_x)
Substituting the values:
θ = arctan(u * sin(θ) / u * cos(θ))
θ = arctan(sin(θ) / cos(θ))
Using the given values for u = 100 m/s, we can substitute them into the equation and then solve for θ using a scientific calculator or a trigonometric table.
3. Range (R):
The range is the horizontal distance covered by the ball while in the air.
- We can find the range using the equation:
R = u_x * T
Substituting the values:
R = (u * cos(θ)) * (2 * (100 / 9.8))
To calculate the time of flight, angle of projection, and range of a ball thrown with a speed of 100m/s and attaining a height of 150m, we can use the equations of projectile motion.
1. Time of Flight:
The time of flight represents the total time the ball spends in the air. It can be calculated using the equation for vertical motion:
h = u*t*sin(theta) - (1/2)*g*t^2 ,
where:
h = maximum height (150 m),
u = initial velocity (100 m/s),
theta = angle of projection,
g = acceleration due to gravity (9.8 m/s^2),
t = time of flight.
Since we are given the values of h, u, and g, we can rearrange the equation to solve for t:
150 = 100*t*sin(theta) - (1/2)*9.8*t^2 .
Next, we need the value of sin(theta). To find it, we can use the equation for horizontal motion:
Range = u * t * cos(theta),
where Range is the horizontal distance covered by the ball.
Given that the Range is the maximum distance covered by the ball, it occurs when the ball returns to the same level from where it was projected. At this point, the height would be zero.
Using the equation for vertical motion again:
0 = u*t*sin(theta) - (1/2)*g*t^2 .
From this equation, we can solve for t:
u*t*sin(theta) = (1/2)*g*t^2 ,
u*sin(theta) = (1/2)*g*t ,
sin(theta) = (1/2)*g*t / u .
Substituting the given values of u, g, and t, we can solve for sin(theta):
sin(theta) = (1/2)*9.8*t / 100 .
Now, we have the value of sin(theta) and can substitute it back into the equation for the time of flight:
150 = 100*t*(1/2)*9.8*t / 100 - (1/2)*9.8*t^2 .
Simplifying the equation will give us the value of t, which represents the time of flight.
2. Angle of Projection:
To find the angle of projection, theta, we can use the equation we solved for in the previous step:
sin(theta) = (1/2)*9.8*t / 100 .
Therefore, we can substitute the value of t we calculated into this equation to find the angle of projection.
3. Range:
The range represents the horizontal distance covered by the ball. It can be calculated using the equation for horizontal motion:
Range = u * t * cos(theta) ,
where u, t, and theta are the initial velocity, time of flight, and angle of projection, respectively.
Once we have the values of t and theta, we can substitute them into this equation to find the range.