In a circus stunt that went horribly wrong the “splat brothers” below inadvertently fired their cannons simultaneously but at different angles (30° and 60°). 146 Fortunately there were no drag forces acting on the day and both cannons fired the performers at the same speed. Unfortunately the brother fired at 60° is twice as heavy as the brother fired at 30°. Who lands first? The heavy brother The lighter brother They both land at the same time
The correct answer and explanation is :
The correct answer is: The lighter brother lands first.
Explanation:
In projectile motion, the time of flight of an object is determined by the vertical component of its velocity and the acceleration due to gravity. Since air resistance is ignored, the only force acting on the performers after they are launched is gravity.
Let’s define the parameters:
- Both brothers are launched at the same speed ( v_0 ).
- The launch angles are 30° and 60°.
- The heavier brother has twice the mass of the lighter brother.
- Gravity acts downward with an acceleration of ( g ).
The time of flight of a projectile is given by:
[
T = \frac{2 v_0 \sin \theta}{g}
]
where:
- ( v_0 ) is the launch speed,
- ( \theta ) is the launch angle,
- ( g ) is acceleration due to gravity.
Step 1: Calculate Time of Flight for Each Brother
For the lighter brother (30° launch angle):
[
T_{30} = \frac{2 v_0 \sin 30^\circ}{g}
]
Since ( \sin 30^\circ = 0.5 ):
[
T_{30} = \frac{2 v_0 (0.5)}{g} = \frac{v_0}{g}
]
For the heavier brother (60° launch angle):
[
T_{60} = \frac{2 v_0 \sin 60^\circ}{g}
]
Since ( \sin 60^\circ = 0.866 ):
[
T_{60} = \frac{2 v_0 (0.866)}{g} = \frac{1.732 v_0}{g}
]
Step 2: Compare the Times
Clearly,
[
T_{60} > T_{30}
]
which means the heavier brother stays in the air longer.
Step 3: Mass Dependency
Mass does not affect the time of flight in projectile motion (as long as there’s no air resistance). Gravity acts equally on all objects regardless of mass.
Thus, since the brother launched at 60° stays in the air longer, the lighter brother (30°) lands first.
Image Representation:
Here is an illustration depicting the two human cannonball performers being launched at different angles, showing their trajectories and flight paths. The lighter brother (30°) lands first, while the heavier brother (60°) stays in the air longer.