To calculate the initial velocity and acceleration required to launch a ball of radius 0.3 m and mass 50 kg from the Earth's surface to a 100 km orbit, we need to consider the gravitational force and the concept of escape velocity.
- Calculate the required escape velocity: The escape velocity is the minimum velocity needed to escape the gravitational pull of the Earth. It can be calculated using the formula: v_escape = sqrt((2 * G * M) / R) where G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2), M is the mass of the Earth (approximately 5.972 × 10^24 kg), and R is the distance from the center of the Earth to the launch point (Earth's radius plus the altitude of the orbit).
R = radius of the Earth + altitude of the orbit = 6,371 km + 100 km = 6,471 km = 6,471,000 m
Substituting the values into the escape velocity formula: v_escape = sqrt((2 * 6.67430 × 10^-11 * 5.972 × 10^24) / 6,471,000) ≈ 11,186 m/s
Determine the initial velocity required: The initial velocity needed to lift off the ball must be greater than or equal to the escape velocity. Therefore, the initial velocity should be at least 11,186 m/s.
Calculate the acceleration at liftoff: At liftoff, the ball experiences two main forces: its weight (mg) acting downward and the upward thrust force (T). The net force acting on the ball is the difference between these forces: F_net = T - mg
To overcome the ball's weight, the upward thrust force must be equal to or greater than the weight. Therefore, the acceleration at liftoff (a_liftoff) can be calculated using Newton's second law: F_net = ma_liftoff
Setting the thrust force equal to the weight: T = mg
Substituting this into the net force equation: F_net = mg - mg = 0
Therefore, at liftoff, the net force is zero, resulting in zero acceleration. The ball doesn't experience any acceleration once it leaves the ground, assuming no additional forces are acting on it.
In summary:
- The initial velocity required to launch the ball to a 100 km orbit is at least 11,186 m/s.
- The acceleration at liftoff is 0 m/s².