A ball is thrown upwards. Neglecting air resistance, what initial upward speed does the ball need to remain in the air for a total time of 10 seconds?
about 60 m/s
about 50 m/s
about 100 m/s
about 110 m/s
about 80 m/s
The correct answer and explanation is :
To determine the initial speed required for a ball to stay in the air for a total time of 10 seconds, we use kinematic equations. The key concept here is that the ball’s motion consists of two equal parts:
- Ascending phase (from launch to the highest point).
- Descending phase (from the highest point back to the ground).
Step 1: Understanding Time of Flight
The total time in the air is 10 seconds, meaning the ball takes 5 seconds to reach its peak and another 5 seconds to fall back down.
Step 2: Using the Kinematic Equation
For an object moving under gravity, the time to reach the highest point is given by:
[
t = \frac{v_0}{g}
]
where:
- ( v_0 ) is the initial velocity (m/s),
- ( g ) is the acceleration due to gravity (9.81 m/s²),
- ( t ) is the time to reach the highest point (5 seconds).
Rearranging the equation for ( v_0 ):
[
v_0 = g \cdot t
]
Substituting the values:
[
v_0 = (9.81 \, \text{m/s}^2) \times (5 \, \text{s})
]
[
v_0 \approx 49.05 \, \text{m/s}
]
Step 3: Choosing the Correct Answer
Among the options given, the closest value is about 50 m/s.
Conclusion
The correct answer is: about 50 m/s.
This means that if a ball is thrown straight up with an initial speed of about 50 m/s, it will reach its highest point in 5 seconds and return to the ground in another 5 seconds, staying in the air for a total of 10 seconds.
