Suppose that an arrow was fired vertically upward from a crossbow at ground level and that it struck the ground 20 seconds later. If air resistance may be neglected, find the initial velocity of the bolt and the maxim altitude that it reached.
height = -4.9t^2 + vt+ c
but it said "groundlevel" ---> c=0
height = -4.9t^2 + vt
0 = -4.9(400) + 20v
20v = 1960
v = 98
So the initial velocity was 98 m/s
height = -4.9t^2 + 98t
d(height)/dt = -9.8t + 98
= 0 at maximum height
-9.8t + 98 = 0
t = 10 seconds
(We should have known that since the time to up and come down was 20 seconds)
at t=10
height = -4.9(100) + 10(98) = 490 m <-- max height
To solve this problem, we can use the basic equations of motion for vertical motion. The key equations we need to use are:
1. v = u + at
2. s = ut + (1/2)at^2
3. v^2 = u^2 + 2as
In these equations:
- v represents the final velocity
- u represents the initial velocity
- a represents the acceleration
- t represents the time taken
- s represents the displacement
Now, let's break down the problem step by step:
Step 1: Determine the acceleration
Since the arrow is fired vertically upward, the only force acting on it is gravity. Therefore, the acceleration can be taken as the acceleration due to gravity, which is approximately -9.8 m/s^2 (negative because it acts in the opposite direction).
Step 2: Determine the final velocity
When the arrow reaches the maximum altitude, its velocity becomes zero. Therefore, v = 0 m/s.
Step 3: Use equation (1) to find the initial velocity
Using v = u + at and substituting the known values, we have:
0 = u - 9.8 * 20
Rearranging the equation, we get:
u = 9.8 * 20
Step 4: Calculate the maximum altitude
To find the maximum altitude, we need to determine the time taken for the arrow to reach it. At maximum altitude, the velocity is zero, so we can use equation (3) to solve for time:
0^2 = u^2 + 2 * (-9.8) * s_max
Rearranging the equation, we get:
s_max = (u^2) / (2 * 9.8)
Substituting the value of u calculated in Step 3, we can solve for s_max.
Step 5: Calculate the initial velocity
Using the equation u = 9.8 * 20, we can calculate the initial velocity.
So, to summarize:
- The initial velocity of the bolt is approximately 196 m/s (calculated in Step 3).
- The maximum altitude the bolt reached is approximately 1964 m (calculated in Step 4).