A 77 g arrow is fired from a bow whose string exerts an average force of 95 N on the arrow over a distance of 81 cm. What is the speed of the arrow as it leaves the bow?
Mass m = 77 g = 0.077 kg
Force F = 125 N
Distance s = 81 cm = 0.81 m
Speed V = ?
We know that
Work done on the arrow = F *s
=>1/2*m*V^2 =F*s
=>V = √((2*F*s)/m) = √((2*125*0.81)/(.077)) = 51.28 m/s
Well, if I were an arrow leaving a bow, I would definitely be in a hurry to get out of there! Now, let's calculate your speed. To do that, we can use the work-energy principle.
The work done on an object is equal to the change in its kinetic energy. This can be written as:
Work = ΔK.E.
Now, the average force exerted on the arrow is equal to the work done on it, given by:
Work = Force x Distance
Plugging in the values, we have:
95 N x 0.81 m = ΔK.E.
Now, since we're looking for the speed, we need to use the formula for kinetic energy:
K.E. = (1/2)mv^2
We know the mass of the arrow is 0.077 kg. So, we can rewrite our equation as:
95 N x 0.81 m = (1/2) x 0.077 kg x v^2
Solving for v, we get:
v^2 = (95 N x 0.81 m x 2) / 0.077 kg
v^2 = 1669.61 m^2/s^2
Finally, taking the square root of both sides, we find:
v ≈ 40.86 m/s
So, the speed of the arrow as it leaves the bow is approximately 40.86 m/s. That's pretty fast! Just make sure to duck if you see it coming your way.
To find the speed of the arrow as it leaves the bow, we can use the work-energy principle.
Step 1: Convert the given mass of the arrow to kilograms.
Given mass of the arrow = 77 g = 77/1000 = 0.077 kg
Step 2: Convert the given distance to meters.
Given distance = 81 cm = 81/100 = 0.81 m
Step 3: Calculate the work done on the arrow.
Work = force x distance
Work = 95 N x 0.81 m
Work = 76.95 N∙m or J (Joules)
Step 4: Use the work-energy principle to find the final kinetic energy.
The work done on an object is equal to the change in its kinetic energy.
Work = ΔKE
Therefore, ΔKE = 76.95 J
Step 5: Find the final kinetic energy.
The kinetic energy of an object can be calculated using the formula:
KE = (1/2)mv^2
ΔKE = KE_final - KE_initial
Since the arrow starts from rest, the initial kinetic energy is zero.
Therefore, KE_final = ΔKE
KE_final = 76.95 J
Step 6: Substitute the values in the kinetic energy formula and solve for v (speed).
KE = (1/2)mv^2
76.95 = (1/2)(0.077)(v^2)
76.95 = 0.0385v^2
v^2 = 76.95 / 0.0385
v^2 = 1993.51
v = √1993.51
v ≈ 44.63 m/s
The speed of the arrow as it leaves the bow is approximately 44.63 m/s.
To find the speed of the arrow as it leaves the bow, we can use the concept of work done. The work done is equal to the change in kinetic energy. Therefore, we need to calculate the work done on the arrow and equate it to the change in kinetic energy.
The formula for work done is:
Work (W) = Force (F) × Distance (d) × cos(θ)
In this case, the force and distance are given. The angle between the force and the distance is not given, but we can assume that the force acts in the same direction as the displacement, so the angle is 0 degrees and cos(0) = 1.
Plugging in the values, we have:
Work (W) = 95 N × 0.81 m × 1
Now, considering the work done on the arrow is equal to the change in kinetic energy, we can write:
Work (W) = ΔKE
Since the arrow starts from rest, the initial kinetic energy (KE) is 0. Therefore, we can write:
W = KE_final - KE_initial
W = KE_final - 0
W = KE_final
So, we substitute W in the equation:
KE_final = 95 N × 0.81 m
Now, we can use the formula for kinetic energy:
KE = 1/2 × mass × velocity^2
Since the mass of the arrow is given as 77 g, we need to convert it to kilograms by dividing by 1000:
Mass = 77 g ÷ 1000 = 0.077 kg
Plugging in the values, we have:
95 N × 0.81 m = 0.5 × 0.077 kg × velocity^2
Now, we can solve for velocity:
velocity^2 = (95 N × 0.81 m) ÷ (0.5 × 0.077 kg)
velocity^2 = 1243.506 kg·m^2/s^2
Taking the square root of both sides, we find:
velocity = √(1243.506 kg·m^2/s^2)
Using a calculator, we get:
velocity ≈ 35.26 m/s
Therefore, the speed of the arrow as it leaves the bow is approximately 35.26 m/s.