An arrow is shot at an angle of θ=45∘ above the horizontal. The
arrow hits a tree a horizontal distance D=220m away, at the same
height above the ground as it was shot. Use g=9.8m/s2 for the
magnitude of the acceleration due to gravity.
Part A
Find ta, the time that the arrow spends in the air.
Part B
How long after the arrow was shot should the apple be dropped,
in order for the arrow to pierce the apple as the arrow hits the
tree?
Answer
General guidance
Initially, calculate the time that the arrow spends in the air. After that, calculate the time of descent. Finally, calculate the time taken for apple to fall from the tree.
The expression for the horizontal range of a projectile which is projected with some angle above the ground can be calculated using the following formula:
Here, 
is the initial velocity, g is the acceleration due to gravity, and 
is the angle of projection.
Rearrange the equation for
.
The time of flight of projectile which is projected with some angle above the ground can be calculated using the following formula:
Step-by-step
Step 1 of 2
(a)
Substitute 220 m for D, 
for g, and 
for 
in the equation.
Substitute 46.43 m/s for V, for g, and for in the equation
.
.
The time that the arrow spends in the air is 6.7 s.
The time of flight of a projectile is equal to the twice the initial vertical component of the velocity divided by the acceleration due to gravity. The term
in the equationrepresents the vertical component of the initial velocity.
Step 2 of 2
(b)
The useful kinematic equation of motion that is the relation between distance travelled (s), initial velocity (u), acceleration, and t time is,
Substitute 0 for
, g for a, and h for s in the above equation and solve for t.
Substitute 6 m for h and for g in the above equation.
The required time is,
The time taken to leave the apple from the tree is 5.59 s.
The time taken to leave the apple from the tree is 5.59 s.
The time taken by the object to reach the ground from the maximum height is known as time of descent. The required time is equal to the time taken to fall the apple minus the time of flight of the arrow.
Answer
The time that the arrow spends in the air is 6.7 s.
The time taken to leave the apple from the tree is 5.59 s.
Answer only
The time that the arrow spends in the air is 6.7 s.
The time taken to leave the apple from the tree is 5.59 s.