You push a skateboard so that it rolls down the road at a speed of 1.30 m/s. You run after the skateboard and jump off the ground with speed 3.80 m/s and an angle of 22.0° above the horizontal hoping to land on the skateboard. How much distance do you need between you and the skateboard to land on it?
To determine the distance you need between you and the skateboard to land on it, we can break down the problem into horizontal and vertical components.
1. Horizontal Component:
The horizontal component of your jump's velocity is given by v_x = v * cos(theta), where v is your jump's speed (3.80 m/s) and theta is the angle of your jump (22.0° above the horizontal). Using this equation, we can calculate the horizontal component:
v_x = 3.80 m/s * cos(22.0°) = 3.52 m/s (rounded to two decimal places)
2. Vertical Component:
The vertical component of your jump's velocity is given by v_y = v * sin(theta), where v and theta are the same as above. Using this equation, we can calculate the vertical component:
v_y = 3.80 m/s * sin(22.0°) = 1.44 m/s (rounded to two decimal places)
3. Time of Flight:
The time it takes for you to land on the skateboard will depend on the vertical component of your jump's velocity. We can solve for the time of flight using the equation d_y = v_y * t + 0.5 * g * t^2, where d_y is the vertical distance traveled (0.0 m), v_y is the vertical component of your jump's velocity (1.44 m/s), t is the time of flight, and g is the acceleration due to gravity (9.8 m/s^2).
0.0 m = 1.44 m/s * t + 0.5 * 9.8 m/s^2 * t^2
Simplifying and solving for t using quadratic formula:
4.9 t^2 + 1.44 t = 0
t(4.9 t + 1.44) = 0
t = 0 (time at the peak) or t = -1.44 / 4.9 (negative root, ignore) or t ≈ 0.29 s (rounded to two decimal places)
4. Horizontal Distance:
The horizontal distance you need to cover to land on the skateboard is the product of the horizontal component of your jump's velocity and the time of flight. Therefore:
d_x = v_x * t
d_x = 3.52 m/s * 0.29 s ≈ 1.02 m (rounded to two decimal places)
Therefore, you need approximately 1.02 meters of distance between you and the skateboard to land on it.
To find the distance you need between you and the skateboard to land on it, we first need to determine the time it takes for you to reach the same horizontal position as the skateboard.
1. Start by breaking down your initial velocity into horizontal and vertical components:
- Horizontal component: vh = 3.80 m/s * cos(22.0°)
- Vertical component: vv = 3.80 m/s * sin(22.0°)
2. Now, calculate the time it takes for you to reach the same horizontal position as the skateboard. Since the horizontal distances traveled by both you and the skateboard will be the same, the time it takes for you to land on the skateboard will be equal to the time it takes for the skateboard to travel a certain distance.
Let's assume the distance between your starting position and the landing point on the skateboard is d. The time it takes for the skateboard to reach that point can be found using the equation of motion:
d = vx * t, where vx is the horizontal velocity of the skateboard.
Rearranging this equation, we have:
t = d / vx
3. Now, let's calculate the time using the horizontal component of the skateboard's velocity:
t = d / (1.30 m/s)
4. Since we already determined the time it takes for you to reach that position in step 2, let's equate both times:
t = d / (1.30 m/s) = distance you travel horizontally / vh
Rearranging this equation, we have:
d = (distance you travel horizontally / vh) * (1.30 m/s)
5. The distance you travel horizontally can be found using your velocity and time:
distance you travel horizontally = vh * t = vh * (distance you travel horizontally / vh) * (1.30 m/s)
Rearranging this equation, we get:
distance you travel horizontally = (distance you travel horizontally) * (1.30 m/s)
6. Now, we can cancel out the distance you travel horizontally on both sides of the equation:
1 = 1.30 m/s
7. Solving for the distance you need to travel horizontally to land on the skateboard:
distance you travel horizontally = 1 / 1.30 m/s
distance you travel horizontally = 0.769 m
Therefore, you need at least 0.769 meters of distance between you and the skateboard to land on it.