For time 𝑡> 0 , the position of an object moving in the xy-plane is given by the parametric equations 𝑥(𝑡) =𝑡𝑐𝑜𝑠(𝑡/2) and 𝑦(𝑡)=𝑡2+2𝑡 . What is the speed of the object at time 𝑡= 1 ?Responses1.1551.1551.3191.3191.3391.3391.810
Question
For time 𝑡> 0 , the position of an object moving in the xy-plane is given by the parametric equations 𝑥(𝑡) =𝑡𝑐𝑜𝑠(𝑡/2) and 𝑦(𝑡)=𝑡2+2𝑡 . What is the speed of the object at time 𝑡= 1 ?Responses1.1551.1551.3191.3191.3391.3391.810
Solution
To find the speed of the object at time t=1, we first need to find the velocity of the object. The velocity is the derivative of the position with respect to time.
First, let's find the derivative of x(t) and y(t) with respect to time t.
The derivative of x(t) = t*cos(t/2) with respect to t is given by the product rule and chain rule of differentiation. The derivative of t is 1 and the derivative of cos(t/2) is -sin(t/2)1/2. So, the derivative of x(t) is dx/dt = cos(t/2) - tsin(t/2)*1/2.
Similarly, the derivative of y(t) = t^2 + 2t with respect to t is dy/dt = 2t + 2.
Now, the speed of the object at time t is given by the magnitude of the velocity vector, which is sqrt[(dx/dt)^2 + (dy/dt)^2].
Substitute t=1 into dx/dt and dy/dt, we get dx/dt = cos(1/2) - 1*sin(1/2)1/2 = 0.877582 - 0.479426 = 0.398156 and dy/dt = 21 + 2 = 4.
So, the speed at t=1 is sqrt[(0.398156)^2 + 4^2] = sqrt[0.158534 + 16] = sqrt[16.158534] = 4.018.
So, the speed of the object at time t=1 is approximately 4.018. This is not one of the options given in the question. Please check the question again.
Similar Questions
2. A particle moving in the xy-plane has position (𝑥(𝑡),𝑦(𝑡)) at time 𝑡≥0, where 𝑑𝑥𝑑𝑡=cos(𝑡2) and 𝑑𝑦𝑑𝑡=𝑒𝑡sin(𝑡2). At time 𝑡=0, the particle is at position (1,2). The figure above shows the path of the particle for 0≤𝑡≤2.(a) Find the position of the particle at time 𝑡=2.
At time 𝑡≥ 0 , a particle moving in the xy-plane has velocity vector given by 𝑣(𝑡)=〈3,2−𝑡2〉 . If the particle is at the point (1,12) at time 𝑡= 0 , how far is the particle from the origin at time 𝑡= 1 ?Responses2.3042.3043.1073.1074.2094.2095.310
For 𝑡≥ 0 , the velocity of a particle moving along the x-axis is given by 𝑣(𝑡) =𝑡3− 6𝑡2+ 10𝑡− 4 . At what time t does the direction of motion of the particle change from right to left?Responses0.5860.5861.1841.1842.0002.0002.816
𝑎(𝑡)=8.7𝑡2+6,where 𝑡 is measured in seconds.1. Write an equation for the velocity 𝑣 at time 𝑡. Simplify your answer.𝑣(𝑡)=
A point in a mechanism has an initial displacement of 1.3cm, and has a velocity given by𝑣(𝑡)=1.6𝑡+2,where 𝑡 is measured in seconds.1. Write an equation for the displacement 𝑠 at time 𝑡.
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.