You push a skateboard so that it rolls down the road at a speed of 1.30 m/s. You run after the skateboard at a speed of 4.00 m/s and while still behind the skateboard you jump off the ground at an angle of 33.0° above the horizontal hoping to land on the skateboard. How much distance do you need between you and the skateboard to jump and land on it?
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To determine the distance between you and the skateboard required to successfully jump and land on it, we need to analyze the horizontal motion of both you and the skateboard.
Let's break down the problem step by step:
Step 1: Analyze the horizontal motion of the skateboard
Since the skateboard is rolling down the road at a constant speed, its horizontal velocity remains constant. The speed of the skateboard is given as 1.30 m/s.
Step 2: Determine the horizontal distance covered by the skateboard while you are in the air
While you are in the air, the skateboard keeps rolling forward. The time you spend in the air depends on your jump. We need to calculate the horizontal distance the skateboard covers during this interval.
Step 3: Calculate the distance between you and the skateboard required for a successful jump
To land on the skateboard, you need to jump such that the horizontal distance you cover matches the distance covered by the skateboard during the time you are in the air.
Now, let's proceed with the calculations:
Step 1: Analyze the horizontal motion of the skateboard
The horizontal velocity of the skateboard remains constant throughout, which is 1.30 m/s.
Step 2: Determine the horizontal distance covered by the skateboard while you are in the air
The time you spend in the air can be calculated using the horizontal distance formula: d = vt, where d is the distance, v is the velocity, and t is the time.
To determine the time you spend in the air, we need to calculate the time it takes for you to reach the maximum height of your jump. This can be done using the vertical motion equations, but since the problem only asks for the horizontal distance, we can skip this step.
Step 3: Calculate the distance between you and the skateboard required for a successful jump
To compute the distance you need to jump, we need to determine the horizontal distance traveled by the skateboard during the time you are in the air.
We can use the formula: distance = velocity × time
The time can be calculated as the horizontal distance you will travel divided by your velocity (4.00 m/s) since the time will be the same for both you and the skateboard.
distance = velocity × time
distance = 1.30 m/s × (distance / 4.00 m/s)
Simplifying the equation:
1.30 m/s × distance = 4.00 m/s × time
Dividing both sides of the equation by 1.30 m/s:
distance = (4.00 m/s × time) / 1.30 m/s
We already know that time = distance / velocity from Step 2, so substituting this into the equation:
distance = (4.00 m/s × (distance / 4.00 m/s)) / 1.30 m/s
Simplifying further:
distance = distance / 1.30
Multiply both sides of the equation by 1.30 to isolate the distance:
1.30 × distance = distance
Subtract distance from both sides of the equation:
0.30 × distance = 0
Since any value of distance multiplied by 0 results in 0, there is no unique solution for this equation. This means that you would have to be positioned right next to the skateboard to jump and land on it.
In conclusion, you need to be very close to the skateboard, ideally right next to it, to successfully jump and land on it.